Using the Laws of Logarithms

Simplify the following expression to obtain an algebraic expression made up only of |\log 2|, |\log 3|, |\log 5,| and constants. ||\log 25 + \log 24 + \log \frac{1}{4} - \log 6 + \log 8 + \log 10 + \log 9|| Step 1

Note that the sixth term is equal to |1|, since the logarithm of |c| in base |c| is equal to |1|. So, ||\log 25 + \log 24 + \log \frac{1}{4} - \log 6 + \log 8 + 1 + \log 9||
Step 2

Using the quotient law of logarithms, simplify the third term. ||\log \frac{1}{4} = \log 1 - \log 4|| Since |\log 1 = 0|, the expression becomes: ||\log 25 + \log 24 - \log 4 - \log 6 + \log 8 + 1 + \log 9||
Step 3

The numbers |25|, |4|, |8|, and |9|, present in the first, third, fifth, and seventh terms, can be represented using a base raised to an exponent. So: ||\log 5^2 + \log 24 - \log 2^2 - \log 6 + \log 2^3 + 1 + \log 3^2||
Step 4

Using the power law of logarithms, put the exponents in front of each term. ||2\log 5 + \log 24 - 2\log 2 - \log 6 + 3\log 2 + 1 + 2\log 3||
Step 5

Using the product law of logarithms, decompose the argument of the second term (|24|) and the fourth term (|6|). ||2\log 5 + \log (3\times 2^3) - 2\log 2 - \log (2\times 3) + 3\log 2 + 1 + 2\log 3||
||2\log 5 + \log 3 + 3\log 2 - 2\log 2 - \log 2 - \log 3 + 3\log 2 + 1 + 2\log 3||
Step 6

Group together like terms. ||2\log 5 + 2\log 3 + 3\log 2 + 1||
This is the desired answer, but it is not the only possible way of rewriting the expression.

Simplify the following expression: |3\ln x + 4\ln x - 2\ln x^3.|

Step 1

Use the power law for logarithms to rewrite the expression. The expression becomes: ||\ln x^3 + \ln x^4 - \ln x^{3\times 2}||
Step 2

Moving through the expression from left to right, use the product and quotient laws of logarithms. The expression becomes:
||\begin{align} \ln x^3 + \ln x^4 - \ln x^6 & = \ln x^3x^4 - \ln x^6\\ \ & = \ln x^7 - \ln x^6\\ \ & = \ln \frac{x^7}{x^6}\\ \ & = \ln x^1 \\ & = \ln x \end{align}||

What is the value of |x| in the following equation? ||3\ 245 = 2\ 500 (1.056)^{4x}||
||\begin{align}
\dfrac{3\ 245}{\color{red}{2\ 500}} &= \dfrac{2\ 500}{\color{red}{2\ 500}} (1.056)^{4x} && \text{inverse operation} \\\\
1.298 &\ = 1.056^{4x} \\\\
\log_{1.056} 1.298 &\ = 4x && \text{definition of log} \\\\
\dfrac{\log_{10} 1.298}{\log_{10} 1.056} &\ = 4x && \text{change of base} \\\\
\dfrac{4.787}{\color{red}{4}} &\approx \dfrac{4x}{\color{red}{4}} && \text{inverse operation} \\\\
1.197 &\approx x \end{align}||

What is the value of |x| in the following equation? ||2 \log_4 \ x - \log_4 \ (16x) = \log _4 \ 9 + 1||
||\begin{align}
2 \log_4 x - \log_4 (16x) &= \log_4 9 + 1 \\
2 \log_4 x - (\log_4 16 + \log_4 x) &= \log_4 9 + 1 && \text{log of a product} \\
2 \log_4 x - (2 + \log_4 x) &= \log_4 9 + 1 && \text{calculating the log} \\
2 \log_4 x - 2 - \log_4 x &= \log _4 9 + 1 && \text{distributive property} \\
\underbrace{2 \log_4 x - \log_4 x} - 2 &= \log_4 9 + 1 && \text{like terms} \\
\log_4 x -2 \color{red}{+2} &= \log_4 9 +1 \color{red}{+2} && \text{inverse operation} \\
\log_4 x \color{red}{- \log_4 9} &= \log_4 9 \color{red}{- \log_4 9} + 3 && \text{inverse operation} \\
\log_4 \left(\dfrac{x}{9}\right) &= 3 && \text{log of a quotient} \\
4^3 &= \dfrac{x}{9} && \text{definition of log} \\
64 \color{red}{\times 9} &= \dfrac{x}{9} \color{red}{\times 9} && \text{inverse operation} \\
576 &= x \end{align}||