Lesson Explainer: Logarithmic Equations with Like Bases Mathematics

In this explainer, we will learn how to solve logarithmic equations with like bases using the laws of exponents and logarithms.

Since the logarithm function to the base 𝑏 is inverse to the exponential function 𝑏, we will also use exponential functions to solve such equations. In essence, by the definition of log𝑥, it follows that given the number 𝐾 to solve log𝑥=𝐾, we must have 𝑏=𝑏𝑥=𝑏.logor

The first steps are usually to simplify the equations using the laws of logarithms. For example, so solve loglog𝑥+3=3 it is not clear what log3 is, so we recall that the sum of logarithms to a fixed base is just the log of a product, and rewrite the left as log(3𝑥)=3 so that, exponentiating both sides to the 6th power, 3𝑥=6=216 and 𝑥=72.

Example 1: Solving Logarithmic Equations

Find 𝑥 such that log(4𝑥4)=2.

Answer

Since log means log, the statement that log(4𝑥4)=2 is the same as 4𝑥4=10=1004𝑥=104𝑥=1044=26.

We must sometimes be wary of extraneous solutions.

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

What is the solution set of the equation log64=2?

Answer

Here, the unknown is the base. Again, we rewrite this as an exponential equation 64=(𝑥+2), which becomes a quadratic: 𝑥+4𝑥+4=64𝑥+4𝑥60=0(𝑥+10)(𝑥6)=0 such that the solution set is apparently {6,10}.

But we are expecting 𝑥+2 to be positive since it is a logarithmic base. This is not true for 𝑥=10, which we dismiss as an extraneous solution. Therefore, the solution set is 6.

Example 3: Solving Logarithmic Equations

Find the solution set of 𝑥+𝑥+1=2logloglog in .

Answer

This looks awkward because of the squared terms: there is no simplification of either 𝑥log or 2log. So, we consider this all as an equation in 𝑦=𝑥log, from which we get 𝑥+𝑥+1=2,logloglog so 𝑥+2𝑥+1=2𝑦+2𝑦+1=2(𝑦+1)=2.logloglogloglog This equation of squares is solved by 𝑦+1=2𝑦+1=2.logorlog Rewriting these two equations using log𝑥 once more, loglog𝑥+1=2 implies that logloglog𝑥2=1𝑥2=1𝑥2=7=17𝑥=27 or loglog𝑥+1=2 implies that logloglog𝑥+2=1(2𝑥)=12𝑥=7=17𝑥=114. The solution set is 27,114.

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