Question

Write each expression as a single logarithm. \(\log _{4}\left(x^{2}-1\right)-5 \log _{4}(x+1)\)

Short answer

\log_{4}((x^{2} - 1)(x + 1)^{5})

Step-by-step solution

Apply the power rule of logarithms

Use the power rule of logarithms to rewrite the term: \(-5 \log_{4}(x+1) = \log_{4}((x+1)^{-5})\).

Substitute the rewritten term back into the expression

Replace \(-5 \log_{4}(x+1)\) in the original expression with \( \log_{4}((x+1)^{-5}) \). The expression becomes: \( \log_{4}(x^{2}-1) - \log_{4}((x+1)^{-5}) \).

Apply the quotient rule of logarithms

Use the quotient rule of logarithms to combine the logs into a single logarithm: \( \log_{4}(x^{2}-1) - \log_{4}((x+1)^{-5}) = \log_{4}\left(\frac{x^{2}-1}{(x+1)^{-5}}\right) = \log_{4}((x^{2} - 1)(x + 1)^{5}) \).

Key concepts

Power Rule of Logarithms

The power rule of logarithms is a useful property when dealing with exponents inside a logarithm. It states that for any positive number \(a\), a logarithm in the form of \(\text{log}_b (a^n)\) can be rewritten as \((n \cdot \text{log}_b (a))\).
This is what it looks like: \(  \text{log}_b (a^n) = n \cdot \text{log}_b (a) \).
Applied to our problem, we have to transform \(-5 \log_{4}(x+1)\) using the power rule.
According to the power rule: \( -5 \log_{4}(x+1) = \log_{4}((x+1)^{-5}) \).
This reduces the complexity of the logarithmic expression and makes it easier to combine with other logarithms later in the solution.

Quotient Rule of Logarithms

The quotient rule of logarithms helps simplify the difference between two logarithms. When subtracting logs of the same base, it allows us to turn the subtraction into a division.
This rule is expressed as: \( \text{log}_b (a) - \text{log}_b (c) = \text{log}_b \left( \frac{a}{c} \right) \).
In our problem, after applying the power rule, our expression is \(\text{log}_{4}(x^{2}-1) - \text{log}_{4}((x+1)^{-5}) \).
Using the quotient rule, this becomes: \( \text{log}_4 \left( \frac{x^{2}-1}{(x+1)^{-5}} \right) \).
This rule lets us combine the logs into a simpler, single logarithmic expression.

Combining Logarithms

Combining logarithms often uses the properties already mentioned, like the power rule and quotient rule, to simplify expressions. After applying these rules, we often end up with a single, more manageable logarithm.
In our example, we previously simplified: \( \text{log}_4 \left( \frac{x^{2}-1}{(x+1)^{-5}} \right) = \text{log}_4 ((x^{2} - 1) \cdot (x + 1)^{5}) \).
Here, we used the fact that dividing by a negative exponent is equivalent to multiplying with the positive exponent inside the logarithm.

Here's a breakdown of the full simplification process:

• First, we applied the power rule to change \(-5 \log_{4}(x+1)\) into \(\text{log}_{4}((x+1)^{-5}) \).
• Then, we used the quotient rule to combine the two logs into one: \(\text{log}_{4} \left( \frac{x^{2}-1}{(x+1)^{-5}} \right) \).
• Finally, we simplified the expression by combining the terms inside the logarithm: \(\text{log}_{4}((x^{2} - 1) \cdot (x + 1)^{5}) \).

Combining logarithms helps to reduce the initial complex expression into a single, easy-to-understand logarithm, making calculations and problem-solving much more straightforward.