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- Exponents and Logarithms

## How to Solve Algebraic Problems With Exponents

Last Updated: May 2, 2023 References

## Solving a Problem with Exponents

## Understanding the Laws of Exponents

## Community Q&A

## Video . By using this service, some information may be shared with YouTube.

## Things You'll Need

- ↑ https://www.mathsisfun.com/operation-order-pemdas.html
- ↑ https://content.nroc.org/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U11_L1_T2_text_final.html
- ↑ https://www.mathsisfun.com/exponent.html
- ↑ http://www.mathwarehouse.com/algebra/exponents/index.php
- ↑ http://www.mathwarehouse.com/algebra/exponents/laws-of-exponents.php
- ↑ http://www.mathsisfun.com/algebra/negative-exponents.html

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## OpenAlgebra.com

## Solving Exponential Equations

## How to Solve an Exponential Equation

How to solve an exponential equation: video lesson, solving an exponential equation: example 1.

Step 1. Take logs of both sides

5 x = 13 becomes log(5 x ) = log(13).

Step 2. Bring down the power in front of the log

Step 3. Solve the resulting equation for x

xlog(5) = log(13) can be solved for x by dividing both sides of the equation by log(5).

This can be evaluated on a calculator to get 𝑥 ≈ 1.59.

## Solving an Exponential Equation: Example 2

3 2x = 0.51 becomes log(3 2x ) = log(0.51).

2xlog(3) = log(0.51) can be solved by dividing both sides by 2log(3).

This can be evaluated as 𝑥 ≈ -0.306.

## How to Solve Exponential Equations with Different Bases

- Take logarithms of both sides of the equation.
- Bring down the exponent in front of the logs.
- Expand and collect x terms.
- Factorise and solve for x.

For example solve the exponential equation 5 𝑥 = 2 𝑥+2 .

Step 1. Take logarithms of both sides

Write each side of the equation inside a log.

5 𝑥 = 2 𝑥+2 becomes log(5 𝑥 ) = log(2 𝑥+2 ).

Step 2. Bring down the exponent in front of the logs

Use the log law log(a b ) = blog(a) to bring the powers down in front of the logs on both sides.

log(5 𝑥 ) = log(2 𝑥+2 ) becomes 𝑥log(5) = (𝑥+2)log2.

Step 3. Expand and collect 𝑥 terms

Expanding the bracket on the right hand side of the equation, (𝑥+2)log2 becomes 𝑥log2 +2log2.

Now the bracket is expanded, the equation becomes 𝑥log5 = 𝑥log2 + 2log.

Step 4. Factorise and solve for 𝑥

Now the 𝑥 terms are on one side of the equation, we can factorise 𝑥 out.

𝑥log5 – 𝑥log2 becomes 𝑥(log5 – log2).

Therefore the equation becomes 𝑥(log5-log2) = 2log2.

To solve for 𝑥, we simply divide both sides of the equation by (log5-log2).

This is the exact answer but it can be evaluated on a calculator as 𝑥 ≈ 1.51.

The image below shows a summary of how to solve the exponential equation in steps.

## Solving Exponential Equations with e

The natural logarithm, ln(x) is the inverse function to e x .

Taking the natural logarithm of both sides, we get:

Here, ln and e cancel out so that:

Now we solve the resulting equation for x by dividing both sides by 2.

This is the exact answer and evaluating it on a calculator, we get x ≈ 0.805.

Solve the exponential equation 5e 3x = 31.

When solving exponential equations with e, the natural logarithm ln(x) is used.

However it is important to first remove any coefficients from in front of the e.

We first divide both sides by 5 to get:

Now the coefficient has been removed, the natural logarithm can be taken.

Now we can solve for x by dividing both sides by 3.

Evaluating this on a calculator 𝑥 ≈ 0.608.

## How to Solve Exponential Equation with the Same Base

## Solving Exponential Equations with the Same Base: Example 1

## Solving Exponential Equations with the Same Base: Example 2

In this example, the bases are already equal. The bases are both 5.

Therefore the exponents must be equal to each other.

Solve this equation for x by adding x to both sides.

Solve for x by dividing both sides of the equation by 3.

## Solving Exponential Equations with the Same Base: Example 3

Firstly, write both sides of the equation with the same base.

We can use 9 = 3 2 and √3 = 3 1 / 2 to help us.

Now the bases are equal, the exponents must be equal.

Now we solve for 𝑥. We add 2 to both sides:

## How to Solve Exponential Equations with Fractional Bases

## Solving an Exponential Equation with a Fractional Base: Example 1

Comparing both sides of the equation, 4 and 8 are both powers of 2.

Now that the bases are equal, the exponents can be equated.

This can be solved for 𝑥 by dividing both sides by -2.

## Solving an Exponential Equation with a Fractional Base: Example 2

In this example, we can compare the fractions on each base.

Now that the bases are the same, the exponents can be equated.

This equation can be solved by first subtracting 𝑥 from both sides to get 6 = 8𝑥.

## Solving an Exponential Equation with a Fractional Base: Example 3

Evaluating this on a calculator, 𝑥 = -5.57.

## How to Solve Exponential Equations Involving Quadratics

The steps for solving an exponential equation involving a quadratic are:

- Write one base as the other base squared
- Substitute this base for k to form a quadratic
- Solve the resulting quadratic for k
- Solve the exponential equation for x

## Solving an Exponential Equation Leading to a Quadratic: Example 1

We know that this equation will lead to a quadratic because 9 x is the square of 3 x .

Step 1. Write one base as the other base squared

9 x = (3 2 ) x which can be written as (3 x ) 2 .

Step 2. Substitute this base for k to form a quadratic

Step 3. Solve the resulting quadratic for k

Step 4. Solve the exponential equations for x

Since we set k = 3 x , if k = 3 or k = 2 then 3 x = 3 or 3 x = 2.

If 3 x = 2 then we can solve this exponential equation with logs.

log(3 x )=log(2) and so, xlog(3) = log(2).

## Solving an Exponential Equation Leading to a Quadratic: Example 2

Step 1. Write one base as the other squared

Since k = 5 x , the solutions of k = 4 and k = 1 become 5 x = 4 and 5 x = 1.

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## Course: Algebra 2 > Unit 6

- Solving exponential equations using exponent properties
- Solving exponential equations using exponent properties (advanced)
- Rational exponents and radicals: FAQ

## Solve exponential equations using exponent properties

- Your answer should be
- an integer, like 6 6 6 6
- a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
- a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
- a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4
- an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
- a multiple of pi, like 12 pi 12\ \text{pi} 1 2 pi 12, space, start text, p, i, end text or 2 / 3 pi 2/3\ \text{pi} 2 / 3 pi 2, slash, 3, space, start text, p, i, end text

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## Solving Exponential Equations with Logarithms

From the Definition With Logarithms With Calculators

## MathHelp.com

## Solve 2 x = 30

log b ( m n ) = n · log b ( m )

Any log of the log's base returns a value of 1 , so log 2 (2) = 1 . Then:

## Solve 5 x = 212 . Give your answer in exact form and as a decimal approximation to three places.

...or about 3.328 , rounded to three decimal places.

## Solve 10 2 x = 52

...or about 0.858 , rounded to three decimal places.

## Solve 3(2 x +4 ) = 350

...or about 2.866 , rounded to three decimal places.

Note: You could also solve the above by using exponent rules to break apart the power on the 2 :

2 x +4 = (2 x )(2 4 ) = (2 x )(16)

Then divide through by the 16 and simplify to get:

Then take the log of each side. You'll get an answer in the form:

URL: https://www.purplemath.com/modules/simpexpo2.htm

## Standardized Test Prep

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## Module 12: Exponential and Logarithmic Equations and Models

Exponential equations, learning outcomes.

- Solve an exponential equation with a common base.
- Rewrite an exponential equation so all terms have a common base then solve.
- Recognize when an exponential equation does not have a solution.
- Use logarithms to solve exponential equations.

## A General Note: Using the One-to-One Property of Exponential Functions to Solve Exponential Equations

For any algebraic expressions S and T , and any positive real number [latex]b\ne 1[/latex],

[latex]{b}^{S}={b}^{T}\text{ if and only if }S=T[/latex]

## How To: Given an exponential equation Of the form [latex]{b}^{S}={b}^{T}[/latex], where S and T are algebraic expressions with an unknown, solve for the unknown

- Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{S}={b}^{T}[/latex].
- Use the one-to-one property to set the exponents equal to each other.
- Solve the resulting equation, S = T , for the unknown.

## Example: Solving an Exponential Equation with a Common Base

Solve [latex]{2}^{x - 1}={2}^{2x - 4}[/latex].

Solve [latex]{5}^{2x}={5}^{3x+2}[/latex].

## Rewriting Equations So All Powers Have the Same Base

## How To: Given an exponential equation with unlike bases, use the one-to-one property to solve it

## Example: Solving Equations by Rewriting Them to Have a Common Base

Solve [latex]{8}^{x+2}={16}^{x+1}[/latex].

Solve [latex]{5}^{2x}={25}^{3x+2}[/latex].

## Example: Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

Solve [latex]{2}^{5x}=\sqrt{2}[/latex].

Solve [latex]{5}^{x}=\sqrt{5}[/latex].

## Example: Determine When an Equation has No Solution

Solve [latex]{3}^{x+1}=-2[/latex].

## Analysis of the Solution

Solve [latex]{2}^{x}=-100[/latex].

## Using Logarithms to Solve Exponential Equations

## How To: Given an exponential equation Where a common base cannot be found, solve for the unknown

- If one of the terms in the equation has base 10, use the common logarithm.
- If none of the terms in the equation has base 10, use the natural logarithm.
- Use the rules of logarithms to solve for the unknown.

## Example: Solving an Equation Containing Powers of Different Bases

Solve [latex]{5}^{x+2}={4}^{x}[/latex].

Solve [latex]{2}^{x}={3}^{x+1}[/latex].

[latex]x=\frac{\mathrm{ln}3}{\mathrm{ln}\left(\frac{2}{3}\right)}[/latex]

Is there any way to solve [latex]{2}^{x}={3}^{x}[/latex]?

## Equations Containing [latex]e[/latex]

## How To: Given an equation of the form [latex]y=A{e}^{kt}[/latex], solve for [latex]t[/latex]

- Divide both sides of the equation by A .
- Apply the natural logarithm to both sides of the equation.
- Divide both sides of the equation by k .

## TIPS for Success

## Example: Solve an Equation of the Form [latex]y=A{e}^{kt}[/latex]

Solve [latex]100=20{e}^{2t}[/latex].

Solve [latex]3{e}^{0.5t}=11[/latex].

Does every equation of the form [latex]y=A{e}^{kt}[/latex] have a solution?

## Example: Solving an Equation That Can Be Simplified to the Form [latex]y=A{e}^{kt}[/latex]

Solve [latex]4{e}^{2x}+5=12[/latex].

Solve [latex]3+{e}^{2t}=7{e}^{2t}[/latex].

[latex]t=\mathrm{ln}\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{2}\mathrm{ln}\left(2\right)[/latex]

## Extraneous Solutions

## Example: Solving Exponential Functions in Quadratic Form

Solve [latex]{e}^{2x}-{e}^{x}=56[/latex].

Solve [latex]{e}^{2x}={e}^{x}+2[/latex].

Does every logarithmic equation have a solution?

## Contribute!

- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- College Algebra. Authored by : Abramson, Jay et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
- Question ID 2637, 2620, 2638. Authored by : Langkamp,Greg. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
- Question ID 98554, 98555, 98596. Authored by : Jenck,Michael. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL

## Exponential Equations

Exponential equations , as the name suggests, involve exponents. We know that the exponent of a number (base) indicates the number of times the number (base) is multiplied. But, what happens if the power of a number is a variable? When the power is a variable and if it is a part of an equation, then it is called an exponential equation. We may need to use the connection between the exponents and logarithms to solve the exponential equations.

Let us learn the definition of exponential equations along with the process of solving them when the bases are the same and when the bases are not the same along with a few solved examples and practice questions.

## What are Exponential Equations?

An exponential equation is an equation with exponents where the exponent (or) a part of the exponent is a variable . For example, 3 x = 81, 5 x - 3 = 625, 6 2y - 7 = 121, etc are some examples of exponential equations. We may come across the use of exponential equations when we are solving the problems of algebra, compound interest , exponential growth , exponential decay , etc.

## Types of Exponential Equations

There are three types of exponential equations. They are as follows:

- Equations with the same bases on both sides. (Example: 4 x = 4 2 )
- Equations with different bases that can be made the same. (Example: 4 x = 16 which can be written as 4 x = 4 2 )
- Equations with different bases that cannot be made the same. (Example: 4 x = 15)

## Equations with Exponents

The equations in algebra involving variable exponents are called equations with exponents or exponential equations. In other words, we can say that algebraic equations in which variables occur as exponents are known as the equations with exponents. Some of the examples of such an equation are, 3 x + 4 = 81, -2 3y-7 = -64, etc.

## Exponential Equations Formulas

While solving an exponential equation , the bases on both sides may be the same or may not be the same. Here are the formulas that are used in each of these cases, which we will learn in detail in the upcoming sections.

## Property of Equality for Exponential Equations

This property is useful to solve an exponential equation with the same bases. It says when the bases on both sides of an exponential equation are equal, then the exponents must also be equal. i.e.,

a x = a y ⇔ x = y.

## Exponential Equations to Logarithmic Form

We know that logarithms are nothing but exponents and vice versa. Hence an exponential equation can be converted into a logarithmic function . This helps in the process of solving an exponential equation with different bases. Here is the formula to convert exponential equations into logarithmic equations.

b x = a ⇔ log b a = x

## Solving Exponential Equations With Same Bases

Sometimes, an exponential equation may have the same bases on both sides of the equation. For example, 5 x = 5 3 has the same base 5 on both sides. Sometimes, though the exponents on both sides are not the same, they can be made the same. For example, 5 x = 125. Though it doesn't have the same bases on both sides of the equation, they can be made the same by writing it as 5 x = 5 3 (as 125 = 5 3 ). To solve the exponential equations in each of these cases, we just apply the property of equality of exponential equations, using which, we set the exponents to be the same and solve for the variable.

Here is another example where the bases are not the same but can be made the same.

Example: Solve the exponential equation 7 y + 1 = 343 y .

We know that 343 = 7 3 . Using this, the given equation can be written as,

7 y + 1 = (7 3 ) y

7 y + 1 = 7 3y

Now the bases on both sides are the same. So we can set the exponents to be the same.

Subtracting y from both sides,

Dividing both sides by 2,

## Solving Exponential Equations With Different Bases

Sometimes, the bases on both sides of an exponential equation may not be the same (or) cannot be made the same. We solve the exponential equations using logarithms when the bases are not the same on both sides of the equation. For example, 5 x = 3 neither has the same bases on both sides nor the bases can be made the same. In such cases, we can do one of the following things.

- Convert the exponential equation into the logarithmic form using the formula b x = a ⇔ log b a = x and solve for the variable.
- Apply logarithm (log) on both sides of the equation and solve for the variable. In this case, we will have to use a property of logarithm , log a m = m log a.

We will solve the equation 5 x = 3 in each of these methods.

We will convert 5 x = 3 into logarithmic form. Then we get,

log 5 3 = x

Using the change of base property ,

x = (log 3) / (log 5)

We will apply log on both sides of 5 x = 3. Then we get, log 5 x = log 3. Using the property log a m = m log a on the left side of the equation, we get, x log 5 = log 3. Dividing both sides by log 5,

Important Notes on Exponential Equations:

Here are some important notes with respect to the exponential equations.

- To solve the exponential equations of the same bases, just set the exponents equal.
- To solve the exponential equations of different bases, apply logarithm on both sides.
- The exponential equations with the same bases also can be solved using logarithms.
- If an exponential equation has 1 on any one side, then we can write it as 1 = a 0 , for any 'a'. For example, to solve 5 x = 1, we can write it as 5 x = 5 0 , then we get x = 0.
- To solve an exponential equation using logarithms, we can either apply "log" or apply "ln" on both sides.

Related Articles:

- Exponential Form
- Exponent Rules
- Exponential Functions
- Exponential Equation Calculator

## Exponential Equations Examples

Example 1: Solve 27 / (3 -x ) = 3 6 .

We know that 27 = 3 3 . We can make the bases to be the same on both sides using this.

3 3 / (3 -x ) = 3 6

Using the quotient property of exponents, a m /a n = a m - n . Using this,

3 3 - (-x) = 3 6

3 3 + x = 3 6

Now the bases on both sides are the same. So we can set the exponents to be equal.

Subtracting 3 from both sides,

Therefore, the solution of the given exponential equation is x = 3.

Example 2: Solve the exponential equation 7 3x + 7 = 490.

490 cannot be written as a power of 7. So we cannot make the bases to be the same here. So we solve this exponential equation using logarithms.

Apply log on both sides of the given equation,

log 7 3x + 7 = log 490

Using a property of logarithms , log a m = m log a. Using this,

(3x + 7) log 7 = log 490 ... (1)

Here, 490 = 49 × 10 = 7 2 × 10.

So, log 490 = log (7 2 × 10)

= log 7 2 + log 10 (because log (mn) = log m + log n)

= 2 log 7 + 1 (because log a m = m log a and log 10 = 1)

Substituting this in (1),

(3x + 7) log 7 = 2 log 7 + 1

Dividing both sides by log 7,

3x + 7 = (2 log 7 + 1) / (log 7)

3x + 7 = 2 + (1 / log 7)

Subtracting 7 from both sides,

3x = -5 + (1 / log 7)

Dividing both sides by 3,

x = -5/3 + (1 / (3 log 7))

Answer: The solution of the given exponential equation is x = -5/3 + (1 / (3 log 7)).

Example 3: How long does it take for $20,000 to double if the amount is compounded annually at 8% annual interest? Round your answer to the nearest integer.

The principal amount, P = $20000.

The rate of interest is, r = 8% = 8/100 = 0.08.

The final amount is, A = 20000 x 2 = $40,000

Let us assume that the required time in years is t.

Using the compound interest formula when compounded annually,

A = P (1 + r) t

40000 = 20000 (1 + 0.08) t

Dividing both sides by 20000,

2 = (1.08) t

Taking log on both sides,

log 2 = log (1.08) t

log 2 = t log (1.08)

t = (log 2) / (log 1.08)

The final answer is rounded to the nearest integer.

Answer: It takes 9 years for $20,000 to double itself.

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## Exponential Equations Questions

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## FAQs on Exponential Equations

What are exponential equations.

An exponential equation is an equation that has a variable in its exponent(s). For example, 5 2x - 3 = 125, 3 7 - 2x = 91, etc are exponential equations.

## What Are Types of Exponential Equations?

There are three types of exponential equations. They are,

- The exponential equations with the same bases on both sides.
- The exponential equations with different bases on both sides that can be made the same.
- The exponential equations with different bases on both sides that cannot be made the same.

## How To Solve Exponential Equations?

To solve the exponential equations of equal bases, we set the exponents equal whereas to solve the exponential equations of different bases, we apply logarithms on both sides.

## How To Write Exponential Equation in Logarithmic Form?

Writing the exponential equation in the logarithmic form helps us to solve it. This can be done using the formula b x = a ⇔ log b a = x.

## What Is the Property of Equality of Exponential Equations?

The equality property of exponential equations says to set the exponents equal whenever the bases on both sides of the equation are equal. i.e., a x = a y ⇔ x = y.

## How To Solve Exponential Equations of Same bases?

When an exponential equation has the same bases on both sides, just set the exponents equal and solve for the variable. Here is an example, 4 2x - 1 = 4 1 - x . Here the bases on both sides are equal. So we can set the exponents equal. 2x - 1 = 1 - x 3x = 2 x = 2/3.

## How To Solve Exponential Equations of Different bases?

When an exponential equation has different bases on both sides, apply log on both sides and solve for the variable. Here is an example, 4 x - 5 = 8. Taking log on both sides, log 4 x - 5 = log 8 (x - 5) log 4 = log 8 x - 5 = (log 8) / (log 4) x = [(log 8) / (log 4)] + 5.

## How To Solve Exponential Equations Using Logarithms?

We solve exponential equations using logarithms in two ways.

- Convert the exponential equation into logarithmic equation using b x = a ⇔ log b a = x.
- Apply "log" or "ln" on both sides and solve.

## IMAGES

## VIDEO

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