# 9.8 – Solving Exponential Equations

## Key Terms

• No key terms for this section.

## Review

Remember…
• The inverse of an exponent is a logarithm, and vice versa.
• Whatever you do to ONE side of the equation, you MUST do to the OTHER side of the equation to keep the value unchanged.
• So, taking the logarithm of both sides of an equation does not change the value of the equation.

## Notes

How to Solve Exponential Equations in the Form $a\bullet b^{x}=d$
• Step 1: Isolate the exponential expression by dividing both sides by a.
• Sometimes a = 1, so you can skip this step.
• Step 2: Isolate the variable by taking the logarithm, base b, of both sides.
• This makes the log cancel out on one side (see Step 3).
• Step 3: Use the property $log_b(b^{x})=x$ to simplify one side.
• Step 4: If the base is NOT base 10 or e, use the change-of-base formula to rewrite the other side of the equation.
• $(log_c a=\frac{log_b a}{log_b c})$
• Step 5: Evaluate the equation to solve for the variable.
• Round to the two decimal places.

Using the Power Property
• You can solve an equation that has an expression with a base other than 10 or e without using the change-of-base formula.
1. Divide both sides by a.
2. Take the common logarithm of both sides.
3. Use the power property to simplify the left side.
1. Move the x to the front (power law)
4. Isolate the variable.
1. Divide by that log on both sides to isolate x
5. Evaluate.
1. Use a calculator to simplify the numerator and denominator, then solve
• Example with a = 1

• Example with a = 8

## Examples

 Ex 1. What is the first step in solving the equation below? $5^{x}=21$ Answer: Take the log of both sides: $log\; 5^{x}=log\; 21$ Ex 2. What is the solution to the equation below? Round your answer to two decimal places. $e^{x}=7.9$ Setup: take the natural log (ln) of both sides, then use a calculator to solve. Answer: x = 2.07 Ex 3. What is the solution to the equation below? Round your answer to two decimal places. $5^{x}=9.2$ Setup: Take the log base 5 of both sides, then use change-of-base and a calculator to solve. $log_5 5^{x}=log_5 9.2$ $x=\frac{log\; 9.2}{log\; 5}$ Answer: x = 1.38 Ex 4. What is the solution to the equation below? Round your answer to two decimal places. $5\bullet e^{x}=15.76$ Setup: divide both sides by 5, then take the natural log of both sides to solve. Answer: x = 1.15 Ex 5. What is the solution to the equation below? Round your answer to two decimal places. $3\bullet 7^{x}=10.48$ Setup: divide both sides by 3, then take the log base 7 of both sides. Set up change-of-base and solve. You could also use the power law instead of change-of-base. Answer: x = 0.64 Ex 6. What is the solution to the equation below? Round your answer to two decimal places. $2^{3x}=73$ Setup: Take the common log of both sides, then use the power law, then simplify and solve. $log\; 2^{3x}=log\; 73$ $3x\bullet log\; 2=log\; 73$ $3x\bullet \frac{log\; 2}{log\; 2}=\frac{log\; 73}{log\; 2}$ $3x=\frac{1.86}{0.3}$ $3x=6.19$ Answer: x = 2.06 Ex 7. What is the solution to the equation below? Round your answer to two decimal places. $3e^{2x}+12=42$ Setup: Subtract 12 from both sides, then take the natural log of both sides, then use the power law, then simplify and solve. $\frac{3e^{2x}}{3}=\frac{30}{3}$ $e^{2x}=10$ $ln_e\; e^{2x}=ln_e\; 10$ $2x=2.30$ Answer: x = 1.15