Search topics

Search

Express one logarithmic quantity in terms of another

Start a set of 8
Step 1 of 5

What is the key first step when the logarithm base differs from the named relation?

\[p=\log_2\left(x\right),\quad \log_4\left(x\right)\]
Year 12, VCE Units 3 & 4, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Expressing a logarithm in terms of a named quantity

Rewrite every logarithm using the base of the named quantity, apply the logarithm laws, and then replace the original variable logarithm. Keep any outside coefficient until the complete expression has been formed

\[\log_a\left(x\right)=\frac{\log_b\left(x\right)}{\log_b\left(a\right)}\]
  1. Change the base When the target base differs from the named logarithm base, put the logarithm of the original argument over the logarithm of the target base \(\displaystyle \log_a\left(x\right)=\frac{\log_b\left(x\right)}{\log_b\left(a\right)}\)
  2. Apply the power law An exponent on a positive argument becomes a coefficient outside its logarithm \(\displaystyle \log_a\left(x^r\right)=r\log_a\left(x\right)\)
  3. Replace the original variable Use the named relation wherever the original variable logarithm remains \(\displaystyle p=\log_b\left(x\right)\quad\Longrightarrow\quad \log_b\left(x\right)=p\)
  4. Keep outside coefficients A coefficient multiplying a logarithm multiplies every term after the logarithm is rewritten \(\displaystyle k\left(r+s p\right)=kr+ksp\)

Worked example

Let q be the base-3 logarithm of positive x. Express the logarithmic expression below in terms of q

\[2\log_9\left(3^2x^4\right)\]
  1. \(\displaystyle \log_9\left(u\right)=\frac{\log_3\left(u\right)}{\log_3\left(9\right)}=\frac{1}{2}\log_3\left(u\right)\)Change the base from 9 to the base 3 used in the named quantity
  2. \(\displaystyle \log_3\left(3^2x^4\right)=\log_3\left(3^2\right)+\log_3\left(x^4\right)=2+4q\)Use the product and power laws, then replace log₃(x) with q.
  3. \(\displaystyle 2\log_9\left(3^2x^4\right)=2\left(\frac{1}{2}\right)\left(2+4q\right)=2+4q\)Apply the outside coefficient and simplify the complete expression

Common mistakes

Inverting change of base
✕\(\displaystyle \log_4\left(x\right)=\frac{\log_2\left(4\right)}{\log_2\left(x\right)}\)✓\(\displaystyle \log_4\left(x\right)=\frac{\log_2\left(x\right)}{\log_2\left(4\right)}\)
The logarithm of the original argument is the numerator and the logarithm of the target base is the denominator
Not folding an outside coefficient
✕\(\displaystyle 2\log_9\left(3^2x^4\right)=\frac{1}{2}\left(2+4q\right)\)✓\(\displaystyle 2\log_9\left(3^2x^4\right)=2\left(\frac{1}{2}\right)\left(2+4q\right)=2+4q\)
The outside coefficient multiplies the whole changed-base expression, including its constant and q terms
Retaining the original variable
✕\(\displaystyle 2\log_9\left(3^2x^4\right)=2+4\log_3\left(x\right)\)✓\(\displaystyle 2\log_9\left(3^2x^4\right)=2+4q\)
The requested form uses the named quantity q, so replace every occurrence of log₃(x) with q.

Difficulty levels

introRewrites one logarithm or a short sum using integer coefficients
standardCombines two laws or uses change of base with a rational coefficient
challengeCombines logarithm laws, inverse identities or change of base before back-substitution
examSelects a complete exact expression and records its coefficients and argument exponents

Questions about this topic

Why is the target base in the denominator of change of base?+

Change of base divides the logarithm of the argument by the logarithm of the original base, with both logarithms taken in the new base

How does an exponent inside a logarithm change the expression?+

For a positive argument, the power law moves the exponent outside the logarithm as a coefficient

Why must the original variable be replaced at the end?+

The question asks for an expression in the named quantity, so any remaining logarithm of the original variable has not yet been rewritten completely

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 4: Logarithmic functions

In this chapter

Connect exponential and logarithmic formsApply logarithm laws and inverse identitiesEvaluate logarithms and antilogarithmsExpress one logarithmic quantity in terms of anotherUse a formula-bearing logarithmic scaleSolve exponential equations and inequalities exactlySolve exponential and logarithmic equationsSolve literal logarithmic and indicial equationsAnalyse exponential and logarithmic graphsConstruct exponential and logarithmic inverse pairsInterpret and use an exponential modelInterpret and use a logarithmic modelDetermine and use a logarithmic model