Question

Write each expression as a single logarithm. \(3 \log _{5}(3 x+1)-2 \log _{5}(2 x-1)-\log _{5} x\)

Short answer

\(\log _{5}\left(\frac{(3x+1)^3}{(2x-1)^2 \times x}\right)\)

Step-by-step solution

- Apply the logarithm power rule

Use the power rule of logarithms: \(\forall a, b \text{ and } c, (c \times \text{log}_a(b) = \text{log}_a(b^c))\), to transfer the coefficients inside the logarithms. This gives us:\(\log _{5}((3x+1)^3) - \log _{5}((2x-1)^2) - \log _{5}x\).

- Combine the logarithms using the quotient rule

Use the quotient rule of logarithms: \(\forall a, b \text{ and } c, \text{log}_a(b) - \text{log}_a(c) = \text{log}_a(b/c)\), to combine the logarithmic expressions.We start by combining the first two terms:\(\log _{5}((3x+1)^3) - \log _{5}((2x-1)^2) = \log _{5}\left(\frac{(3x+1)^3}{(2x-1)^2}\right)\).

- Combine the result with the third logarithm

Combine the result with the third term using the quotient rule again:\(\log _{5}\left(\frac{(3x+1)^3}{(2x-1)^2}\right) - \log _{5}x = \log _{5}\left(\frac{(3x+1)^3}{(2x-1)^2 \times x}\right)\).

Key concepts

logarithm power rule

The logarithm power rule is a handy tool for simplifying logarithmic expressions. It states that for any positive real numbers \(a\) and \(b\), and any real number \(c\), the expression \(c \times \, \text{log}_a(b)\) can be rewritten as \(\text{log}_a(b^c)\). In simpler terms, you can bring a multiplier in front of a logarithm inside the logarithm as an exponent. This transformation helps reduce the complexity of our problem.
For example, consider the term \(3 \log_5(3x + 1)\). Using the power rule, we can rewrite this as \(\log_5((3x + 1)^3)\). This makes it easier to combine this logarithm with others.
In our exercise, by applying the power rule to each term, we transformed the expression from \(3 \log_5(3x+1) - 2 \log_5(2x-1) - \log_5 x\) to \(\log_5((3x+1)^3) - \log_5((2x-1)^2) - \log_5 x\). This step simplifies how we can deal with each logarithmic term in the following steps.

quotient rule of logarithms

The quotient rule of logarithms is essential for combining logarithms that are subtracted. The rule states: for any positive real numbers \(a\), \(b\), and \(c\), the expression \( \text{log}_a(b) - \text{log}_a(c)\) can be combined into \(\text{log}_a(b/c)\). This rule is useful when dealing with logarithms in subtraction.
In our exercise, we first applied it to combine \(\log_5((3x+1)^3) - \log_5((2x-1)^2)\). According to the quotient rule, these can be combined into \(\log_5\left( \frac{(3x+1)^3}{(2x-1)^2} \right)\).
Next, we needed to integrate the remaining term \(- \log_5 x\). Again using the quotient rule, we combined this with our existing logarithm: \( \log_5\left( \frac{(3x+1)^3}{(2x-1)^2} \right) - \log_5 x\). This further simplifies to \( \log_5\left( \frac{(3x+1)^3}{(2x-1)^2 \times x} \right)\). Using the quotient rule step by step helps in managing the complexity of the expression.

combining logarithms

Combining logarithms is a crucial step to simplify expressions that have multiple logarithms involved. By using rules like the power rule and quotient rule, we can transform and merge different logarithmic terms into a single logarithm. The final step in our example exercise involves combining all the logarithms into one.
After applying the power rule and quotient rule twice, we ended up with the expression \(\log_5\left( \frac{(3x+1)^3}{(2x-1)^2 \times x} \right)\). This final expression is much cleaner and easier to interpret.
By systematically using the rules for logarithms, we can manage even initially complicated expressions. This approach demystifies logarithmic operations and makes it easier to solve logarithmic equations and manipulate logarithmic expressions in algebra.