## How to Rewrite Logarithms

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### Rewriting the Logarithms

If you realize that logarithms are just another way to write out **exponential** equations, it becomes much easier to solve a logarithm. Once you **change** the logarithm into a form that is easier to understand, you should be able to **solve** it the **same** way you would solve an ordinary exponential equation.

### Steps for Rewriting Logarithms

**Step1: **Learn what a logarithm is. Before figuring out how to solve logarithms, you need to know that a logarithm is just a different way to write an exponential equation. Here's a clear explanation of what it means:

- \(y \ = \ log_b \ (x) \ ⇒ \ b^y \ = \ x\)
- Remember that the
**base**of a logarithm is \(b\). Also, the following must be true:

- \(b \ > \ 0\)
- \(b\) does
**not**equal \(1\)

- In the same equation, \(x\) is the exponential expression that the logarithm is set to be equal to, and \(y\) is the exponent.

**Step2: **Examine the equation. Find the base (\(b\)), the exponent (\(y\)), and the exponential expression (\(x\)).

**Example: **\(log_3 \ 729 \ = \ 6\)

- \(b \ = \ 3\)
- \(y \ = \ 6\)
- \(x \ = \ 729\)

**Step3: **Put the exponential expression on one side of the equation. Put the value of your exponential expression, \(x\), on one side of the equal sign.

**Example:** \(729 \ = \ ?\)

**Step4: **Use the exponent to multiply the base. Your exponent, \(y\), tells you how **many times** you need to multiply the value of your base, \(b\), by itself.

**Example:** \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \ = \ ?\)

You could also write this as \(3^6\)

**Step5: **You should now be able to **write** the logarithm as an exponential expression. Make sure your answer is correct by **checking** that both sides of the equation are the **same**.

**Example:** \(3^6 \ = \ 729\)

### Here are some rules you should know if you want to rewrite logarithms:

**Product rule:** The "product rule" is the first property of logarithms. It says that the logarithm of the **product** of two numbers is equal to the **sum** of the logarithms of the two numbers. In the form of a math equation:

- \(log_b \ (m \times n) \ = \ log_b \ (m) \ + \ log_b \ (n)\)
- Note the following must also be true:

- \(m \ > \ 0\)
- \(n \ > \ 0\)

**Quotient rule:** The second property of logarithms, called the "quotient rule," says that the logarithm of a **quotient** can be found by **subtracting** the logarithm of the denominator from the logarithm of the numerator. In the form of a math equation:

- \(log_b \ (\frac{m}{n}) \ = \ log_b \ (m) \ - \ log_b \ (n)\)
- Note the following must also be true:

- \(m \ > \ 0\)
- \(n \ > \ 0\)

### Exercises for Rewriting Logarithms

**1) **Rewrite the logarithm: \(log_6 \ 36 \ = \ 2\)

**2) **Rewrite the logarithm: \(log_{10} \ 10000 \ = \ 4\)

**3) **Rewrite the logarithm: \(log_4 \ 84.4485 \ = \ 3.2\)

**4) **Rewrite the logarithm: \(log_7 \ 59.5258 \ = \ 2.1\)

**5) **Rewrite the logarithm: \(log_{11} \ 1331 \ = \ 3\)

**6) **Rewrite the logarithm: \(log_8 \ 4096 \ = \ 4\)

**7) **Rewrite the logarithm: \(log_8 \ 4096 \ = \ 4\)

**8) **Rewrite the logarithm: \(log_{15} \ 507.0022 \ = \ 2.3\)

**9) **Rewrite the logarithm: \(log_{19} \ 361 \ = \ 2\)

**10) **Rewrite the logarithm: \(log_9 \ 908.1378 \ = \ 3.1\)

**1) **Rewrite the logarithm: \(log_6 \ 36 \ = \ 2\)

\(\color{red}{6^2 \ = \ 36}\)

**2) **Rewrite the logarithm: \(log_{10} \ 10000 \ = \ 4\)

\(\color{red}{10^4 \ = \ 10000}\)

**3) **Rewrite the logarithm: \(log_4 \ 84.4485 \ = \ 3.2\)

\(\color{red}{4^{3.2} \ = \ 84.4485}\)

**4) **Rewrite the logarithm: \(log_7 \ 59.5258 \ = \ 2.1\)

\(\color{red}{7^{2.1} \ = \ 59.5258}\)

**5) **Rewrite the logarithm: \(log_{11} \ 1331 \ = \ 3\)

\(\color{red}{11^3 \ = \ 1331}\)

**6) **Rewrite the logarithm: \(log_8 \ 4096 \ = \ 4\)

\(\color{red}{8^4 \ = \ 4096}\)

**7) **Rewrite the logarithm: \(log_8 \ 4096 \ = \ 4\)

\(\color{red}{8^4 \ = \ 4096}\)

**8) **Rewrite the logarithm: \(log_{15} \ 507.0022 \ = \ 2.3\)

\(\color{red}{15^{2.3} \ = \ 507.0022}\)

**9) **Rewrite the logarithm: \(log_{19} \ 361 \ = \ 2\)

\(\color{red}{19^2 \ = \ 361}\)

**10) **Rewrite the logarithm: \(log_9 \ 908.1378 \ = \ 3.1\)

\(\color{red}{9^{3.1} \ = \ 908.1378}\)