Question

evaluate the following logarithms \\(\log_{4}(256)=\\) \\(\log_{5}(125)=\\) \\(\log_{2}(4)=\\)

Explanation:

Step1: Evaluate $\log_{4}(256)$

Recall the definition of a logarithm: $\log_{b}(a) = x$ means $b^{x}=a$. We need to find $x$ such that $4^{x}=256$. We know that $4^{4}=256$ (since $4\times4 = 16$, $16\times4 = 64$, $64\times4 = 256$), so $\log_{4}(256)=4$.

Step2: Evaluate $\log_{5}(125)$

Using the logarithm definition, find $x$ where $5^{x}=125$. We know that $5^{3}=125$ (since $5\times5 = 25$, $25\times5 = 125$), so $\log_{5}(125)=3$.

Step3: Evaluate $\log_{2}(4)$

Using the logarithm definition, find $x$ where $2^{x}=4$. We know that $2^{2}=4$, so $\log_{2}(4)=2$.

Answer:

$\log_{4}(256)=\boldsymbol{4}$
$\log_{5}(125)=\boldsymbol{3}$
$\log_{2}(4)=\boldsymbol{2}$