Questions: logm m^n

Transcript text: $\log _{m} m^{n}$

Solution

Solution Steps

Solution Approach

To solve the logarithmic expression $\log_{m} m^{n}$, we can use the logarithm power rule, which states that $\log_{b} (a^c) = c \cdot \log_{b} a$. In this case, since the base $m$ and the argument $m^n$ are the same, the expression simplifies to $n$.

Step 1: Identify the Expression

We start with the expression $\log_{m} m^{n}$.

Step 2: Apply the Logarithm Power Rule

Using the logarithm power rule, we have: \[ \log_{b} (a^c) = c \cdot \log_{b} a \] In our case, this simplifies to: \[ \log_{m} m^{n} = n \cdot \log_{m} m \]

Step 3: Simplify the Expression

Since $\log_{m} m = 1$, we can further simplify: \[ \log_{m} m^{n} = n \cdot 1 = n \]

Step 4: Substitute the Value of $n$

Given $n = 3$, we find: \[ \log_{m} m^{n} = 3 \]