Math Worksheet Collection: Evaluating Logarithmic Expressions ... - Free Printable

To solve the given logarithmic problems, we need to use the definition of logarithms and some basic properties. The logarithm $\log_b(a)$ is defined as the exponent to which the base $b$ must be raised to obtain $a$. That is:

\[
\log_b(a) = x \quad \text{if and only if} \quad b^x = a
\]

Let's solve each problem step by step.

Left Column

1. $\log_7(49)$
- We need to find $x$ such that $7^x = 49$.
- Since $49 = 7^2$, we have $x = 2$.
- Therefore, $\log_7(49) = 2$.

2. $\log_2(32)$
- We need to find $x$ such that $2^x = 32$.
- Since $32 = 2^5$, we have $x = 5$.
- Therefore, $\log_2(32) = 5$.

3. $\log_{12}(1)$
- We need to find $x$ such that $12^x = 1$.
- Since any number raised to the power of 0 is 1, we have $x = 0$.
- Therefore, $\log_{12}(1) = 0$.

4. $\log_4\left(\frac{1}{16}\right)$
- We need to find $x$ such that $4^x = \frac{1}{16}$.
- Since $\frac{1}{16} = 4^{-2}$, we have $x = -2$.
- Therefore, $\log_4\left(\frac{1}{16}\right) = -2$.

5. $\log_3(3)$
- We need to find $x$ such that $3^x = 3$.
- Since $3 = 3^1$, we have $x = 1$.
- Therefore, $\log_3(3) = 1$.

6. $\log_4(16)$
- We need to find $x$ such that $4^x = 16$.
- Since $16 = 4^2$, we have $x = 2$.
- Therefore, $\log_4(16) = 2$.

7. $\log_2\left(\frac{1}{2}\right)$
- We need to find $x$ such that $2^x = \frac{1}{2}$.
- Since $\frac{1}{2} = 2^{-1}$, we have $x = -1$.
- Therefore, $\log_2\left(\frac{1}{2}\right) = -1$.

8. $\log_{10}(100)$
- We need to find $x$ such that $10^x = 100$.
- Since $100 = 10^2$, we have $x = 2$.
- Therefore, $\log_{10}(100) = 2$.

9. $\log_4(4)$
- We need to find $x$ such that $4^x = 4$.
- Since $4 = 4^1$, we have $x = 1$.
- Therefore, $\log_4(4) = 1$.

10. $\log_2(2)$
- We need to find $x$ such that $2^x = 2$.
- Since $2 = 2^1$, we have $x = 1$.
- Therefore, $\log_2(2) = 1$.

Right Column

1. $\log_4(1)$
- We need to find $x$ such that $4^x = 1$.
- Since any number raised to the power of 0 is 1, we have $x = 0$.
- Therefore, $\log_4(1) = 0$.

2. $\log_3(81)$
- We need to find $x$ such that $3^x = 81$.
- Since $81 = 3^4$, we have $x = 4$.
- Therefore, $\log_3(81) = 4$.

3. $\log_{10}\left(\frac{1}{1000}\right)$
- We need to find $x$ such that $10^x = \frac{1}{1000}$.
- Since $\frac{1}{1000} = 10^{-3}$, we have $x = -3$.
- Therefore, $\log_{10}\left(\frac{1}{1000}\right) = -3$.

4. $\log_7(7)$
- We need to find $x$ such that $7^x = 7$.
- Since $7 = 7^1$, we have $x = 1$.
- Therefore, $\log_7(7) = 1$.

5. $\log_3(9)$
- We need to find $x$ such that $3^x = 9$.
- Since $9 = 3^2$, we have $x = 2$.
- Therefore, $\log_3(9) = 2$.

6. $\log_7(1)$
- We need to find $x$ such that $7^x = 1$.
- Since any number raised to the power of 0 is 1, we have $x = 0$.
- Therefore, $\log_7(1) = 0$.

7. $\log_{10}\left(\frac{1}{100}\right)$
- We need to find $x$ such that $10^x = \frac{1}{100}$.
- Since $\frac{1}{100} = 10^{-2}$, we have $x = -2$.
- Therefore, $\log_{10}\left(\frac{1}{100}\right) = -2$.

8. $\log_7\left(\frac{1}{7}\right)$
- We need to find $x$ such that $7^x = \frac{1}{7}$.
- Since $\frac{1}{7} = 7^{-1}$, we have $x = -1$.
- Therefore, $\log_7\left(\frac{1}{7}\right) = -1$.

9. $\log_8(1)$
- We need to find $x$ such that $8^x = 1$.
- Since any number raised to the power of 0 is 1, we have $x = 0$.
- Therefore, $\log_8(1) = 0$.

10. $\log_7\left(\frac{1}{49}\right)$
- We need to find $x$ such that $7^x = \frac{1}{49}$.
- Since $\frac{1}{49} = 7^{-2}$, we have $x = -2$.
- Therefore, $\log_7\left(\frac{1}{49}\right) = -2$.

Final Answer

\[
\boxed{
\begin{array}{ll}
\log_7(49) = 2 & \log_4(1) = 0 \\
\log_2(32) = 5 & \log_3(81) = 4 \\
\log_{12}(1) = 0 & \log_{10}\left(\frac{1}{1000}\right) = -3 \\
\log_4\left(\frac{1}{16}\right) = -2 & \log_7(7) = 1 \\
\log_3(3) = 1 & \log_3(9) = 2 \\
\log_4(16) = 2 & \log_7(1) = 0 \\
\log_2\left(\frac{1}{2}\right) = -1 & \log_{10}\left(\frac{1}{100}\right) = -2 \\
\log_{10}(100) = 2 & \log_7\left(\frac{1}{7}\right) = -1 \\
\log_4(4) = 1 & \log_8(1) = 0 \\
\log_2(2) = 1 & \log_7\left(\frac{1}{49}\right) = -2 \\
\end{array}
}
\]