Solving exponential and logarithmic equations
Isolate an exponential expression or combine logarithms before solving for the variable. Apply the same logarithm to both sides, check every original logarithm argument, and round only the final approximation
\[a^{f(x)}=b\Longleftrightarrow f(x)=\log_a(b)\quad(a>0,\;a\ne1,\;b>0)\]
- Isolate the exponential Make the exponential expression the subject before applying logarithms \(\displaystyle 3\times5^{x-1}=18\Rightarrow5^{x-1}=6\)
- Apply the same logarithm Apply one logarithm base to both sides so the exponent can be brought down \(\displaystyle 5^x=12\Rightarrow\log_5(5^x)=\log_5(12)\Rightarrow x=\log_5(12)\)
- Convert a logarithm to exponential form A logarithm states the exponent that produces its positive argument \(\displaystyle \log_3(x+1)=2\Rightarrow x+1=3^2=9\)
- Check domains and round once Every original logarithm argument must be positive, and an approximate answer is rounded only at the end \(\displaystyle \log_2(x-3)=4\Rightarrow x=19,\quad x-3=16>0\)
Worked example
Solve the logarithmic equation exactly and check the domain
\[\log_3\left(x+1\right)+\log_3\left(3\right)=2\]
- \(\displaystyle x+1>0,\quad3>0\)Both logarithm arguments in the original equation must be positive
- \(\displaystyle \log_3\left(3\left(x+1\right)\right)=2\)Use the product law to combine the two same-base logarithms
- \(\displaystyle 3\left(x+1\right)=3^2=9\Rightarrow x+1=3\)Convert to exponential form, then divide by 3.
- \(\displaystyle x=2,\quad2+1=3>0\)The candidate satisfies the original domain, so it is a valid solution
Common mistakes
Taking a logarithm on one side
✕\(\displaystyle 4^{2x+1}=32\Rightarrow\log_{4}\left(4^{2x+1}\right)=32\Rightarrow2x+1=32\)✓\(\displaystyle 4^{2x+1}=32\Rightarrow\log_{4}\left(4^{2x+1}\right)=\log_{4}(32)\Rightarrow2x+1=\log_{4}(32)\)
The same logarithm must be applied to both sides, so the right side becomes log base 4 of 32, not 32.Treating the exponent as a factor
✕\(\displaystyle 5^{2x+1}=20\Rightarrow2x+1=20\div5\)✓\(\displaystyle 5^{2x+1}=20\Rightarrow2x+1=\log_5(20)\)
The exponent is brought down by a logarithm, not by dividing by the baseIgnoring a logarithm domain
✕\(\displaystyle \log_2(x-1)+\log_2(x-3)=3\Rightarrow x=5\;\text{or}\;x=-1\)✓\(\displaystyle \log_2(x-1)+\log_2(x-3)=3\Rightarrow x=5,\quad5-1=4>0,\quad5-3=2>0\)
Every logarithm argument must be positive in the original equation, so negative one is rejectedDifficulty levels
introIsolate one exponential, convert it with a logarithm, and find one solution
standardSolve a two-base or combined-log equation with one admissible exact result
challengeSolve a logarithmic quadratic with an extraneous candidate, or use a graph or enclosing interval
examChoose a method and exact form, then give the final approximate solution over an interval
Questions about this topic
Why must the logarithm be applied to both sides of an exponential equation?+
Applying the same operation to both sides preserves equality. Applying a logarithm to one side only creates a different equation
Why can a logarithmic equation produce an extraneous algebraic solution?+
Algebra can produce a value that does not make every original logarithm argument positive. That value must be rejected after checking the original domains
Why should rounding be delayed until the end?+
Rounding an intermediate logarithm changes the value used in later operations and can change the final requested digit
Can I use a calculator?+
A calculator is allowed unless a question is marked no 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