Solving Logarithms - High School Math

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Question

You are given that $\log_{10} x = A$ and $\log_{10} y = B$ .

Which of the following is equal to $100^{2A-B}$ ?

Answer

Since $\log_{10} x = A$ and $\log_{10} y = B$ , it follows that $10^{A} = x$ and $10^{B} = y$

$100^{2A-B} = \left( 10^{2} \right ) ^{2A-B} =10^{2 (2A-B)} =10^{4A-2B}=\frac{10^{4A}}{10^{2B}} =\frac{\left ( 10^{A}\right )^4}{\left (10^{B}\right )^2}=\frac{x^4}{y^2}$

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Question

$\textup{Solve for }x\textup{: },\log_{2}32=x$

Answer

$\textup{This can be re-written as an exponent problem: } 2^{x}=32$

$2\times2\times2\times2\times2=32:\textup{ so }32=2^{5}:::x=5$

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Question

What is $log_{2}(8)$ ?

Answer

Recall that by definition a logarithm is the inverse of the exponential function. Thus, our logarithm corresponds to the value of in the equation:

$2^{x} = 8$ .

We know that $2^{3} = 8$ and thus our answer is .

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