Question

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where​ possible, evaluate logarithmic expressions.
log left parenthesis 4 x plus 1 right parenthesis minus log left parenthesis x right parenthesis
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Part 1
log left parenthesis 4 x plus 1 right parenthesis minus log left parenthesis x right parenthesisequals
  
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​(Type an exact answer in simplified form. Use integers or fractions for any numbers in the​ expression.)

Answer 1

To condense the logarithmic expression \( \log(4x + 1) - \log(x) \) using the properties of logarithms, we can apply the quotient rule of logarithms, which states that:

\[
\log(a) - \log(b) = \log\left(\frac{a}{b}\right)
\]

In your case, we have \( a = 4x + 1 \) and \( b = x \). Applying the quotient rule, we get:

\[
\log(4x + 1) - \log(x) = \log\left(\frac{4x + 1}{x}\right)
\]

Thus, the condensed form of the expression is:

\[
\log\left(\frac{4x + 1}{x}\right)
\]

That is the final condensed expression. If you would like to simplify further, you can express it as:

\[
\log\left(4 + \frac{1}{x}\right) \quad (\text{after dividing each term in the numerator by } x)
\]

So, the most simplified single logarithm expression will be:

\[
\log\left(4 + \frac{1}{x}\right)
\]

However, if you are looking for the first condensed form without additional simplification, you can simply use:

\[
\log\left(\frac{4x + 1}{x}\right)
\]

Make sure to provide the answer based on the requirement you have about further simplification!