Why Expand a Logarithmic Expression?
A logarithm may contain several factors, a quotient, or powers all inside one argument. Expanding rewrites that expression as a sum or difference of simpler logarithms. This makes the roles of the factors and exponents easier to see, and it is useful when comparing or manipulating logarithmic expressions.
Expansion is not a change in value: it applies established logarithm rules to an expression whose argument is positive. The conditions matter. Each logarithm must have a positive argument, and all logarithms in a calculation must use the same valid base.
The Rules and Their Conditions
Let \(b>0\) and \(b\ne 1\), so that \(b\) is a valid logarithm base. For positive numbers \(u\) and \(v\), the product and quotient rules are
For \(u>0\) and any real number \(r\), the power rule is
These rules are applied to positive quantities. In particular, a variable that appears as a logarithm argument must be positive. When a quantity is raised to a real exponent inside a logarithm, it is safest to state explicitly that the quantity being powered is positive.
Some useful special values are \(\log_b 1=0\) and \(\log_b b=1\), since \(b^0=1\) and \(b^1=b\). Numerical factors can therefore become numerical terms when expanded: for example, \(\log_2 8=3\).
Why the Power Rule Works
The power rule follows directly from the definition of a logarithm as an inverse of an exponential function. Suppose \(u>0\), and set \(y=\log_b u\). By the definition, \(u=b^y\). For any real \(r\), raising both sides to \(r\) gives \(u^r=(b^y)^r=b^{ry}\). Taking the logarithm to base \(b\) of both sides then gives
The positive-base condition ensures that real powers are defined, and \(u>0\) ensures that the logarithm on the left has a positive argument. The product and quotient rules can likewise be understood from the laws of exponents: multiplying powers of the same base adds their exponents, while dividing them subtracts their exponents.
A Reliable Order for Expanding
When an expression has multiple layers, work from its overall structure rather than trying to move every exponent at once. First identify products and quotients. Then apply the power rule to each powered factor, including powers that result from a grouped expression. Finally, evaluate any numerical logarithms you know exactly and simplify the resulting coefficients.
Confirm that the base is positive and not 1, and that every factor inside a logarithm is positive.
Products produce sums; factors in a denominator produce subtracted logarithms.
For a positive quantity \(u\), replace \(\log_b(u^r)\) with \(r\log_b u\).
Evaluate exact numerical logarithms, then verify the result using valid values or the logarithm rules.
Worked Examples
Worked Example: Expanding a Product and Quotient
Expand \(\log_3\left(\frac{81x^2}{y^3}\right)\), assuming \(x>0\) and \(y>0\).
First separate the numerator and denominator with the quotient rule. Then separate the numerator's product and use the power rule:
Here, \(\log_3 81=4\) because \(3^4=81\). The assumptions \(x>0\) and \(y>0\) make all the variable logarithms defined. To check, choose \(x=3\) and \(y=9\). The original argument is \(81\cdot 9/9^3=729/729=1\), so the original logarithm is \(0\). The expanded result is \(4+2(1)-3(2)=4+2-6=0\), as required.
Worked Example: Including Numerical Factors and a Fractional Power
Expand \(\log_2\left(\frac{a^3b^2}{8\sqrt{c}}\right)\), assuming \(a>0\), \(b>0\), and \(c>0\).
Write the square root as a power, separate the quotient and products, and apply the power rule:
The numerical term is \(\log_2 8=3\), since \(2^3=8\). For a check, let \(a=4\), \(b=2\), and \(c=4\). The original argument is \((4^3\cdot 2^2)/(8\sqrt{4})=(64\cdot4)/(8\cdot2)=256/16=16\), so its logarithm is \(\log_2 16=4\). The expanded expression gives \(3(2)+2(1)-3-\frac12(2)=6+2-3-1=4\).
Worked Example: Expanding a Power of a Product
Expand \(\log_3\left(\frac{(x^2y)^{3/2}}{z^4}\right)\), assuming \(x>0\), \(y>0\), and \(z>0\).
Separate the quotient first. The numerator is a power of a product, so apply the power rule to the whole product and then the product rule inside it:
The coefficient on \(\log_3 x\) is \(\frac32\cdot2=3\). To check, take \(x=3\), \(y=9\), and \(z=3\). The numerator is \((3^2\cdot9)^{3/2}=81^{3/2}=(\sqrt{81})^3=9^3=729\); the denominator is \(3^4=81\). The full argument is \(729/81=9\), so the original logarithm is \(\log_3 9=2\). The expanded expression gives \(3(1)+\frac32(2)-4(1)=3+3-4=2\).
Common Mistakes and Their Corrections
A frequent error is to treat every operation inside a logarithm as if it could be separated. The product and quotient rules apply to multiplication and division, not addition and subtraction.
For a concrete counterexample, use base \(10\) and positive values \(u=1\), \(v=9\). The incorrect split gives \(\log_{10}1+\log_{10}9=0+\log_{10}9=\log_{10}9\), which is not \(1\). But \(\log_{10}(1+9)=\log_{10}10=1\). Thus the two expressions are unequal. Keep the sum together unless some separate valid algebraic transformation changes the argument.
Another common mistake is to lose the minus sign for a denominator factor or to forget that an exponent becomes a coefficient. For instance, \(\log_b(u/v^2)\) expands to \(\log_b u-2\log_b v\), not \(\log_b u+2\log_b v\). Writing the quotient rule before applying the power rule helps preserve the sign.
Why the Conditions Matter
Expansion rules preserve the value of the original expression only when their inputs satisfy the logarithm conditions. If a variable occurs as a logarithm argument, it must be positive. In the examples above, positivity was stated before expansion so that expressions such as \(\log_3 x\) and \(\log_2 c\) were defined over the real numbers.
Do not assume that a squared quantity automatically lets you write a logarithm of the unsquared variable without checking its sign. For example, \(\log_b(x^2)=2\log_b x\) requires \(x>0\) on the right-hand side. If \(x<0\), the left-hand side is defined but \(\log_b x\) is not a real logarithm. The stated power rule is used only when its base quantity is positive.
Expansion is useful because it reveals how each factor contributes: numerator factors give positive terms, denominator factors give negative terms, and exponents become coefficients. A quick check of the base, the positivity assumptions, and the signs often catches errors before they spread into later calculations.
Check Your Understanding
Try each question by stating any needed positivity assumptions and showing the rule used at each step.
- Expand \(\log_5(25m^3n)\), assuming \(m>0\) and \(n>0\).
- Expand \(\log_2\left(\frac{p^4}{16q^2}\right)\), assuming \(p>0\) and \(q>0\).
- Expand \(\log_7\big((r s^2)^{1/2}\big)\), assuming \(r>0\) and \(s>0\).
- Explain why \(\log_b(u+v)\) cannot generally be replaced by \(\log_b u+\log_b v\).
- For \(\log_3\left(\frac{9x^2}{y}\right)\), what conditions on \(x\) and \(y\) allow the expansion \(2+2\log_3 x-\log_3 y\)?
