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Solve exponential equations and inequalities exactly

Start a set of 8
Step 5 of 5

Skipping the common base. One of these lines is right. Which one?

Year 12, VCE Units 3 & 4, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Solving exponential equations exactly

Rewrite compatible powers with one admissible base before comparing exponents. For inequalities, the order reverses when the common base is between 0 and 1, and a substitution u = a^x must stay positive

\[a^{f(x)}=a^{g(x)}\Rightarrow f(x)=g(x)\quad(a>0,\;a\ne1)\]
  1. Rewrite to a common base Express both powers using one admissible base before comparing their exponents \(\displaystyle 8^{x+1}=2^{12}\Rightarrow\left(2^{3}\right)^{x+1}=2^{12}\)
  2. Multiply exponents in a power of a power The outer exponent multiplies the inner exponent \(\displaystyle \left(2^{3}\right)^{x+1}=2^{3(x+1)}\)
  3. Respect the base direction For an equation equate exponents; for an inequality reverse the order when the base is between 0 and 1. \(\displaystyle \left(\frac{1}{2}\right)^x<\left(\frac{1}{2}\right)^4\Rightarrow x>4\)
  4. Keep substitutions valid If u = a^x, use only positive values of u and then solve a^x = u for x. \(\displaystyle u=2^x>0,\quad u=4\Rightarrow2^x=4=2^2\Rightarrow x=2\)

Worked example

Solve the exponential equation exactly

\[27^{x-1}=9^{2x+1}\]
  1. \(\displaystyle 27^{x-1}=9^{2x+1}\Rightarrow\left(3^3\right)^{x-1}=\left(3^2\right)^{2x+1}\)Rewrite 27 and 9 as powers of the common base 3
  2. \(\displaystyle 3^{3(x-1)}=3^{2(2x+1)}\Rightarrow3x-3=4x+2\)Multiply the exponents and then equate them
  3. \(\displaystyle 3x-3=4x+2\Rightarrow-5=x\Rightarrow x=-5\)Subtract 3x and subtract 2 to isolate the solution
  4. \(\displaystyle 27^{-6}=9^{-9}=3^{-18}\)At x equals negative five, both original sides equal three to the power of negative eighteen, so the solution checks

Common mistakes

Skipping the common base
✕\(\displaystyle 8^{x+1}=2^{12}\Rightarrow x+1=12\Rightarrow x=11\)✓\(\displaystyle 8^{x+1}=2^{12}\Rightarrow2^{3(x+1)}=2^{12}\Rightarrow3x+3=12\)
The exponents can be equated only after both sides have been written with the same base
Forgetting the decreasing base
✕\(\displaystyle \left(\frac{1}{2}\right)^x<\left(\frac{1}{2}\right)^4\Rightarrow x<4\)✓\(\displaystyle \left(\frac{1}{2}\right)^x<\left(\frac{1}{2}\right)^4\Rightarrow x>4\)
A base between 0 and 1 gives a decreasing exponential, so the exponent inequality reverses
Keeping an invalid substitution root
✕\(\displaystyle u=2^x,\quad u^2-3u-4=0\Rightarrow u=-1\;\text{or}\;u=4\Rightarrow x=-1\;\text{or}\;x=4\)✓\(\displaystyle u=2^x>0,\quad u=4=2^2\Rightarrow x=2\)
The exponential quantity u is positive, and the remaining value of u is four and must be converted back through the original exponential equation

Difficulty levels

introSolve a common-base equation with an integer solution
standardRewrite one base for a rational solution, or solve an inequality with an increasing base
challengeSolve a decreasing-base inequality or use a quadratic substitution with an inadmissible root
examChoose a method or verdict, then give an exact solution or solution set

Questions about this topic

Why must the bases be the same before exponents are equated?+

The one-to-one rule for powers applies to powers of the same positive base that is not 1. Different bases can produce the same value with different exponents

Why does an inequality reverse when the base is between 0 and 1?+

Powers of a base between 0 and 1 decrease as the exponent increases. Therefore a smaller power value corresponds to a larger exponent

Why must an exponential substitution be positive?+

A positive base raised to any real exponent gives a positive value. Zero and negative polynomial roots cannot equal the substituted exponential quantity

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