Explainer: Logarithmic Equations with Different Bases

In this explainer, we will learn how to solve logarithmic equations involving logarithms with different bases.

We begin by recapping the rules associated with changing the base of a logarithm.

Change of Base of Logarithms

Suppose that 𝑎,𝑏 and 𝑐 are positive numbers, with 𝑎≠1 and 𝑏≠1. Then,

  1. logloglog𝑐𝑏=𝑐.
  2. logloglog𝑏⋅𝑐=𝑐.

Notice that the condition on 𝑎 and 𝑏 is just so that the logarithms are defined at all (not possible with base 1).

This result is not completely expected because what logarithms “do” is convert between products and sums, which allows for the simplification of expressions like loglog𝑥+𝑦 and log𝑥𝑦. Note, of course, that the change of base does NOT help with the product loglog𝑥⋅𝑦.

One way of understanding the terminology is that, in statement (2), in computing the logarithm of the number 𝑐, we have changed the base from 𝑏 (in log𝑐) to the base 𝑎 (of the right-hand side log𝑐). The result says that in order to do this, we can just multiply by log𝑎𝑏.

Recall where this comes from: the result for exponents that says that 𝑝=(𝑝).

Using it we have 𝑎=𝑐=𝑏=𝑎=𝑎,loglogloglogloglog⋅ giving 𝑐 as a power of 𝑎 in two ways, which must be the same: log𝑐.

The change of base allows us to evaluate log𝑥 as a logarithm to the base 𝑏, since loglogloglog𝑥=𝑏𝑥=𝑘𝑥, so loglog𝑥=𝑥𝑘.

Although we could use the fact we have just established above, we will solve the following problem directly.

Example 1: Solving Logarithmic Equations by Changing the Base

Find the solution set of loglog𝑥=4 in ℝ.

Answer

We will use the fact that 9=3 and change base from 9 to 3 on the right loglogloglogfromthegivenequationloglogbythepowerlaw4=9⋅4=24()=2𝑥=𝑥().

Now, since we can recover both sides by raising to the power 3, we find 4=𝑥𝑥∈{−2,2}.or

The solution 𝑥=−2 is extraneous, because log(−2) is not defined, so we conclude that the solution set ={2}.

The following example requires a little search for the application of change of base.

Example 2: Solving Logarithmic Equations Where the Unknown Is in the Base

Determine the solution set of the equation loglog𝑥+254=10 in ℝ.

Answer

Base change, assuming that a solution 𝑥>0 (where 𝑥≠1) exists, says that logloglog𝑥⋅4=4=1 so that, if we set 𝑦=𝑥,log the equation becomes 𝑦+251𝑦=10 or 𝑦+25=10𝑦𝑦−10𝑦+25=0(𝑦−5)(𝑦−5)=0.

So, 𝑦=𝑥=5log and 𝑥=4=1,024.

The solution set is just {1,024}.

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