Can I expand $\log(x+y)$?

No. There is no logarithmic property to expand the logarithm of a sum or difference.

What if there are multiple subtracted logarithms, like $\log(a) - \log(b) - \log(c)$?

Factor out the negative sign or apply the quotient rule sequentially. Both yield $\log\left(\frac{a}{bc}\right)$. Negative terms always contribute to the denominator.

What if the bases are different?

You cannot directly condense logarithms with different bases using these rules. You must first use the change-of-base formula to make all bases equal.

How to expand and condense logarithmic expressions

Expanding a logarithmic expression means rewriting it as a sum, difference, or multiple of simpler logarithms. Condensing is the reverse operation, combining a sum or difference of logarithms into a single logarithmic term.

These operations apply whenever you need to simplify an equation to solve for a variable inside a logarithm, or when preparing functions for differentiation in calculus.

The setup

You must memorize the three primary properties of logarithms, which hold for any base $b > 0$ , $b eq 1$ , and positive arguments $x$ and $y$ :

Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
Power Rule: $\log_b(x^n) = n \log_b(x)$

The steps

To expand an expression:

Apply the Quotient Rule to split fractional arguments into subtraction.
Apply the Product Rule to split multiplied terms into addition.
Apply the Power Rule to bring exponents down as coefficients. To condense an expression:
Apply the Power Rule to move coefficients up as exponents.
Group the terms and apply the Product and Quotient rules left-to-right. Positive terms go in the numerator, negative terms in the denominator.

Checking the result

To check your work, perform the reverse operation. If you expanded an expression, condense your final answer to verify it matches the original problem. Ensure all arguments of the logarithms remain strictly positive, as logarithms of negative numbers or zero are undefined.

Common errors

Do not distribute logarithms over addition or subtraction: $\log_b(x + y) eq \log_b(x) + \log_b(y)$ . Do not apply the Power Rule to the entire logarithm: $(\log_b(x))^n eq n \log_b(x)$ .

Worked example

Part A: Expand $\log_2\left(\frac{8x^3}{\sqrt{y}}\right)$ . Part B: Condense $2\ln(x) + \ln(y) - \frac{1}{2}\ln(z)$ .

Part A: Expand Original expression: $\log_2\left(\frac{8x^3}{\sqrt{y}}\right)$ Apply the Quotient Rule: $\log_2(8x^3) - \log_2(\sqrt{y})$ Apply the Product Rule to the first term: $\log_2(8) + \log_2(x^3) - \log_2(y^{1/2})$ Apply the Power Rule and evaluate $\log_2(8) = 3$ : $3 + 3\log_2(x) - \frac{1}{2}\log_2(y)$

Part B: Condense Original expression: $2\ln(x) + \ln(y) - \frac{1}{2}\ln(z)$ Apply the Power Rule: $\ln(x^2) + \ln(y) - \ln(z^{1/2})$ Apply the Product Rule for the addition: $\ln(x^2y) - \ln(\sqrt{z})$ Apply the Quotient Rule for the subtraction: $\ln\left(\frac{x^2y}{\sqrt{z}}\right)$

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References: OpenStax Algebra and Trigonometry, Chapter 6: Exponential and Logarithmic Functions · Khan Academy, Unit: Logarithms