Study Notes on Logarithmic Functions

Solutions to Logarithmic Functions Exam

1. Question: Express the exponential equation $5^3 = 125$ in its equivalent logarithmic form.
   Solution: Using the definition $ext{if } b^y = x, ext{ then } y = ext{log}_b{x}$, we have $b=5$, $y=3$, and $x=125$. Therefore, the logarithmic form is $ext{log}_5{125} = 3$.

2. Question: Solve for $y$ in the equation $y = ext{log}_7{49}$.
   Solution: The equation $y = ext{log}_7{49}$ implies $7^y = 49$. Since we know that $7^2 = 49$, it follows that $y = 2$.

3. Question: Determine the value of $ext{ln}{e^5}$.
   Solution: The natural logarithm $ext{ln}{x}$ is defined as $ext{log}_e{x}$. So, $ext{ln}{e^5}$ is equivalent to $ext{log}_e{e^5}$. If we let $y = ext{log}_e{e^5}$, this means $e^y = e^5$. Therefore, $y = 5$.

4. Question: If $ext{log}_b{81} = 4$, what is the value of $b$?
   Solution: The equation $ext{log}_b{81} = 4$ can be rewritten in exponential form as $b^4 = 81$. To find $b$, we need to determine which base raised to the power of 4 equals 81. We know that $3^4 = 3 \times 3 \times 3 \times 3 = 81$. Thus, $b = 3$.