The Logarithm Power Law is a rule that states that the logarithm of an exponential is equal to the exponent times the logarithm of the base. It can be used to solve equations with unknown exponents. It can be expressed algebraically as:
Evaluate \(\log_264^3\).
We can evaluate the following logarithm by applying the Power Law of Logarithms:
\(\log_264^3 = 3\log_264\)
\(\log_264^3 = 3\log_22^6\)
\(\log_264^3 = 3(6)\)
\(\log_264^3 = 18\)
Therefore, we can evaluate \(\log_264^3\) as equal to \(\boldsymbol{18}\).
\(\log1000^{-3}\)
We can evaluate the following logarithm by applying the Power Law of Logarithms. Since this represents a Natural Logarithm, we will list the base as \(10\):
\(\log_{10}1000^{-3} = -3\log_{10}1000\)
\(\log_{10}1000^{-3} = -3\log_{10}10^3\)
\(\log_{10}1000^{-3} = -3(3)\)
\(\log_{10}1000^{-3} = -9\)
Therefore, we can evaluate \(\log1000^{-3}\) as equal to \(\boldsymbol{-9}\).
\(\log_5\sqrt{125}\)
First, we can rewrite the logarithm as such:
Next, we can evaluate the following logarithm by applying the Power Law of Logarithms:
\(\log_{5}125^{\frac{1}{2}} = \cfrac{1}{2}\log_{5}125\)
\(\log_{5}125^{\frac{1}{2}} = \cfrac{1}{2}\log_{5}5^3\)
\(\log_{5}125^{\frac{1}{2}} = \cfrac{1}{2}(3)\)
\(\log_{5}125^{\frac{1}{2}} = 1.5\)
Therefore, we can evaluate \(\log_5\sqrt{125}\) as equal to \(\boldsymbol{1.5}\).
Any logarithm can be expressed in terms of common logarithms using the change of base formula:
This formula can be used to evaluate logarithms or graph logarithmic functions with any base.
Evaluate \(\log_{4}26\).
First, we can express this logarithm in terms of common logarithms:
Next, we can use a calculator to evaluate the value of the logarithm:
\(\log_{4}26 = \cfrac{1.414973348}{0.602059991}\)
\(\log_{4}26 = 2.350\)
Therefore, we can determine that \(\log_{4}26\) evaluates to \(\boldsymbol{2.350}\).
\(\log_{1/3}42\)
First, we can express this logarithm in terms of common logarithms:
Next, we can use a calculator to evaluate the value of the logarithm:
\(\log_{1/3}42 = \cfrac{1.62324929}{-0.477121254}\)
\(\log_{1/3}42 = -3.402\)
Therefore, we can determine that \(\log_{1/3}42\) evaluates to \(\boldsymbol{-3.402}\).
\(\log_{2.75}97\)
First, we can express this logarithm in terms of common logarithms:
Next, we can use a calculator to evaluate the value of the logarithm:
\(\log_{2.75}97 = \cfrac{1.986771734}{0.439332693}\)
\(\log_{2.75}97 = 4.522\)
Therefore, we can determine that \(\log_{2.75}97\) evaluates to \(\boldsymbol{4.522}\).