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Problem 754

# Express the logarithm of 7 to the base 3 in terms of common logarithms.

Expert verified
The logarithm of 7 to the base 3 in terms of common logarithms is $$\log_3 7 = \frac{\log 7}{\log 3}$$.
See the step by step solution

## Step 1: Identify the given logarithm and the desired base

We are given the logarithm $$\log_3 7$$ and we want to express it in terms of common logarithms, which are base 10.

## Step 2: Apply the change of base formula

Now we will use the change of base formula to convert our given logarithm to base 10. $$\log_{a}b = \frac{\log_{c}b}{\log_{c}a}$$ In our case, $$a = 3$$, $$b = 7$$, and $$c = 10$$. So, we have: $$\log_3 7 = \frac{\log_{10} 7}{\log_{10} 3}$$

## Step 3: Simplify the expression

We can now use the simplified notation for common logarithms, which is $$\log b = \log_{10} b$$. So our expression becomes: $$\log_3 7 = \frac{\log 7}{\log 3}$$

## Step 4: Write the final answer

Now we have expressed the logarithm of 7 to the base 3 in terms of common logarithms: $$\log_3 7 = \frac{\log 7}{\log 3}$$

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