How to Rewrite Logarithms

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:
  
  • \(m \ > \ 0\)
  • \(n \ > \ 0\)

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