## How to Solve Exponential Equations Requiring Logarithms

This article will focus on solving exponential equations that require using logarithms, which are more challenging to answer without them. This is because we can't change the exponential equation to that both sides of the equation have the same base. If you run into this kind of problem, you should do the following:

### Steps for Solving Exponential Equations Using Logarithms

1. Put the exponential expression on one side of the equation by itself.
2. Find the logarithms of both sides of the equation. Logs can be made from any material.
3. Find out what the variable is. You can give an exact answer or a decimal approximation.

Besides the steps above, you should also look over the Basic Logarithm rules because you will use them in some way.

Now let's review some basic logarithm rules quickly, then solve an example

• $$log_b \ (m \times n) \ = \ log_b \ m \ + \ log_b \ n$$
• $$log_b \ (\frac{m}{n} \ ) \ = \ log_b \ m \ - \ log_b \ n$$
• $$log_b \ m^n \ = \ n \ log_b \ m$$
• $$log_a \ m \ = \ \frac{log_b \ m}{log_b \ a}$$
• $$log_n \ m \ = \ \frac{1}{log_m \ n}$$
• $$ln \ 1 \ = \ 0$$
• $$ln \ (e) \ = \ 1$$
• $$ln \ e^x \ = \ x$$
• $$e^{ln \ x} \ = \ x$$
• $$ln \ 0$$ is undefined.

### Solving Exponential Equations $$a \times b^x \ = \ d$$

Let's solve:  $$7 \times 3^x \ = \ 217$$

Before we can solve for $$x$$, we need to put the exponential expression on one side. To do this, divide each side by $$7$$. Notice that we don't multiply the $$7$$ and $$3$$ together because that's not how math works!

$$7 \times 3^x \ = \ 217 \ ⇒ \ 3^x \ = \ 31$$

Now we should convert the equation to logarithmic form.

$$3^x \ = \ 31 \ ⇒ \ log_3 \ 31 \ = \ x$$

Now that the equation is logarithmic, we can figure out what $$x$$ is. The exact answer is $$x \ = \ log_3 \ 31$$

Since $$31$$ is not a rational power of $$3$$, we must use the change of base rule and our calculators to figure out the logarithm.

Change of base rule: $$x \ = \ log_3 \ 31 \ = \ \frac{log \ 31}{log \ 3}$$

Evaluate using a calculator: $$\frac{log \ 31}{log \ 3} \ ≈ \ \frac{1.49136}{0.47712} \ ≈ \ 3.1257$$

### Exercises for Solving Exponential Equations Requiring Logarithms

1) Find the value of $$x$$: $$5^x \ = \ 17$$

2) Find the value of $$z$$: $$9^z \ = \ 30$$

3) Find the value of $$r$$: $$3^{r \ + \ 4} \ = \ 85$$

4) Find the value of $$a$$: $$4^{2a \ - \ 5} \ = \ 67$$

5) Find the value of $$x$$: $$e^x \ = \ 15$$

6) Find the value of $$x$$: $$5 \ e^{x \ - \ 2} \ = \ 25$$

7) Find the value of $$x$$: $$4.5^{x \ + \ 1} \ = \ 46$$

8) Find the value of $$x$$: $$6 \ e^{2b \ - \ 3} \ = \ 154.2$$

9) Find the value of $$r$$: $$7^r \ = \ 158$$

10) Find the value of $$b$$: $$8^b \ + \ 9 \ = \ 231$$

1) Find the value of $$x$$: $$5^x \ = \ 17$$

$$\color{red}{5^x \ = \ 17 \ ⇒ \ x \ ln \ 5 \ = \ ln \ 17 \ ⇒ \ x \ = \ \frac{ln \ 17}{ln \ 5} \ ⇒ \ x \ = \ 1.7603}$$

2) Find the value of $$z$$: $$9^z \ = \ 30$$

$$\color{red}{9^z \ = \ 30 \ ⇒ \ z \ ln \ 9 \ = \ ln \ 30 \ ⇒ \ z \ = \ \frac{ln \ 30}{ln \ 9} \ ⇒ \ z \ = \ 1.5479}$$

3) Find the value of $$r$$: $$3^{r \ + \ 4} \ = \ 85$$

$$\color{red}{3^{r \ + \ 4} \ = \ 85 \ ⇒ \ (r \ + \ 4) \ ln \ 3 \ = \ ln \ 85 \ ⇒}$$$$\color{red}{ \ r \ + \ 4 \ = \ \frac{ln \ 85}{ln \ 3} \ ⇒ \ r \ + \ 4 \ = \ 4.0438 \ ⇒ \ r \ = \ 0.0438}$$

4) Find the value of $$a$$: $$4^{2a \ - \ 5} \ = \ 67$$

$$\color{red}{4^{2a \ - \ 5} \ = \ 67 \ ⇒ \ (2a \ - \ 5) \ ln \ 4 \ = \ ln \ 67 \ ⇒ \ 2a \ - \ 5 \ = \ \frac{ln \ 67}{ln \ 4} \ ⇒}$$ $$\color{red}{2a \ - \ 5 \ = \ 3.033 \ ⇒ \ 2a \ = \ 8.033 \ ⇒ \ a \ = \ 4.016}$$

5) Find the value of $$x$$: $$e^x \ = \ 15$$

$$\color{red}{e^x \ = \ 15 \ ⇒ \ x \ ln \ e \ = \ ln \ 15 \ ⇒ \ x \ = \ ln \ 15 \ = \ 2.708}$$

6) Find the value of $$x$$: $$5 \ e^{x \ - \ 2} \ = \ 25$$

$$\color{red}{5 \ e^{x \ - \ 2} \ = \ 25 \ ⇒ \ (x \ - \ 2) \ ln \ e \ = \ ln \ 5 \ ⇒ \ x \ = \ ln \ 5 \ + \ 2 \ = \ 3.6094}$$

7) Find the value of $$x$$: $$4.5^{x \ + \ 1} \ = \ 46$$

$$\color{red}{4.5^{x \ + \ 1} \ = \ 46 \ ⇒ \ (x \ + \ 1) \ ln \ 4.5 \ = \ ln \ 46 \ ⇒ \ x \ = \ \frac{ln \ 46}{ln \ 4.5} \ - \ 1 \ = \ 1.5455}$$

8) Find the value of $$x$$: $$6 \ e^{2b \ - \ 3} \ = \ 154.2$$

$$\color{red}{6 \ e^{2b \ - \ 3} \ = \ 154.2 \ ⇒ \ (2b \ - \ 3) \ ln \ e \ = \ ln \ 25.7 \ ⇒ \ 2b \ = \ ln \ 25.7 \ + \ 3 \ ⇒ }$$ $$\color{red}{ \ b \ = \ \frac{ln \ 25.7 \ + \ 3}{2} \ = \ 3.1232}$$

9) Find the value of $$r$$: $$7^r \ = \ 158$$

$$\color{red}{7^r \ = \ 158 \ ⇒ \ r \ ln \ 7 \ = \ ln \ 158 \ ⇒ \ r \ = \ \frac{ln \ 158}{ln \ 7} \ ⇒ \ x \ = \ 2.6016}$$

10) Find the value of $$b$$: $$8^b \ + \ 9 \ = \ 231$$

$$\color{red}{8^b \ + \ 9 \ = \ 231 \ ⇒ \ b \ ln \ 8 \ = \ ln \ 222 \ ⇒ \ b \ = \ \frac{ln \ 222}{ln \ 8} \ ⇒ \ b \ = \ 2.5981}$$

## Solving Exponential Equations Requiring Logarithms Practice Quiz

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