Chapter 4.3, Problem 3E

(a)

To determine

To fill: The blank in the equation $5^3=125\text{, so }{\log }_{}\left(\right)=\left(\right)$.

Answer to Problem 3E

The complete equation is $5^3=125\text{, so }\underset{¯}{{\log }_5\left(125\right)=\left(3\right)}$.

Explanation of Solution

Definition used:

Logarithmic function:

Let a be a positive number with $a\ne 1$, then the logarithmic function with base a, denoted by ${\log }_a$ is defined by ${\log }_a\left(x\right)=y$ implies $a^y=x$.

Calculation:

The given exponential equation is $5^3=125$.

Compare the given equation with $a^y=x$ and obtain the constant values as follows.

$a=5,y=3\text{ and }x=125$

Substitute $a=5,y=3\text{ and }x=125$ in ${\log }_a\left(x\right)=y$ and find the exponential form of given equation as shown below.

${\log }_{5}125=4$

Thus, the complete equation is $5^3=125\text{, so }\underset{¯}{{\log }_5\left(125\right)=\left(3\right)}$.

(b)

To determine

To fill: The blank in the equation ${\log }_{5}25=2,{\text{ so \_\_}}^{\_\_\_}=\_\_\_$.

Answer to Problem 3E

The complete equation is ${\log }_{5}25=2,\text{ so }\underset{¯}{{\text{5}}^2=25}$.

Explanation of Solution

The given logarithm equation is ${\log }_{5}25=2$.

Use the definition mentioned in sub part (a) and compare the given equation with ${\log }_a\left(x\right)=y$ as shown below.

$a=5,x=25\text{ and }y=2$

Substitute $a=5,x=25\text{ and }y=2$ in $a^y=x$ as shown below.

$\begin{array}{l}{a}^y=x\\ 5^2=25\end{array}$

Thus, the complete equation is ${\log }_{5}25=2,\text{ so }\underset{¯}{{\text{5}}^2=25}$.