Mechanical Engineering Exam  >  Mechanical Engineering Notes  >  General Aptitude for GATE  >  How to Solve Exponential Equations using Logarithms

How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering PDF Download

Steps to Solve Exponential Equations using Logarithms

  1. Keep the exponential expression by itself on one side of the equation.
  2. Get the logarithms of both sides of the equation. You can use any bases for logs.
  3. Solve for the variable. Keep the answer exact or give decimal approximations. In addition to the steps above, make sure that you review the Basic Logarithm Rules because you will use them in one way or another.
    How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

Examples of How to Solve Exponential Equations using Logarithms 

Example 1: Solve the exponential equation 52x = 21.
Sol
:
The good thing about this equation is that the exponential expression is already isolated on the left side. We can now take the logarithms of both sides of the equation. It doesn’t matter what base of the logarithm to use. The final answer should come out the same. The best choice for the base of log operation is 5 since it is the base of the exponential expression itself. However, we will also use in the calculation the common base of 10, and the natural base of  e (denoted by ln) just to show that in the end, they all have the same answers.
Log Base of 5
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Log Base of 10
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

Log Base of e
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

Example 2: Solve the exponential equation 2 (3x–5)=12. 
Sol: As you can see, the exponential expression on the left is not by itself. We must eliminate the number 2 that is multiplying the exponential expression. To do that, divide both sides by 2. That would leave us just the exponential expression on the left, and 6 on the right after simplification.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
It’s time to take the log of both sides. Since the exponential expression has base 3, that’s the convenient base to use for log operation. In addition, we will also solve this using the natural base e just to compare if our final results agree.

Log Base of 3
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Log Base of e
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

Example 3: Solve the exponential equation How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Sol:

This looks like a mess at first. However, if you know how to start this out, the solution to this problem becomes a breeze. What we should do first is to simplify the expression inside the parenthesis. Use the Division Rule of Exponent by copying the common base of e and subtracting the top by the bottom exponent.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Now isolate the exponential expression by adding both sides by 7, followed by dividing the entire equation by 2.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Take the logarithm of both sides. Use ln because we have a base of e. Then solve for the variable x.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

Example 4: Solve the exponential equation How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Sol:

Observe that the exponential expression is being raised to x. Simplify this by applying the Power to a Power Rule. Do that by copying the base 10 and multiplying its exponent to the outer exponent. It should look like this after doing so.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
We can now isolate the exponential expression by subtracting both sides by 3 and then multiplying both sides by 2.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Take the logarithm of both sides with base 10. If you just see a log without any specific base, it is understood to have 10 as its base.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
We are going to solve this quadratic equation by factoring method. Let’s move everything to the left side, therefore making the right side equal to zero. Factor out the trinomial into two binomials. Set each binomial factor equal zero then solve for x.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

Example 5: Solve the exponential equation e2x –7ex +10 = 0.
Sol:  

We will need a different strategy to solve this exponential equation. Observe that we can actually convert this into a factorable trinomial. First, we let m = ex. Rewrite the exponential expression using this substitution.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Factor out the trinomial as a product of two binomials. Then replace m m by ex again.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering
Finally, set each factor equal to zero and solve for x, as usual, using logarithms.
How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

The document How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering is a part of the Mechanical Engineering Course General Aptitude for GATE.
All you need of Mechanical Engineering at this link: Mechanical Engineering
198 videos|165 docs|152 tests

Top Courses for Mechanical Engineering

198 videos|165 docs|152 tests
Download as PDF
Explore Courses for Mechanical Engineering exam

Top Courses for Mechanical Engineering

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

Summary

,

video lectures

,

past year papers

,

Previous Year Questions with Solutions

,

MCQs

,

Semester Notes

,

Viva Questions

,

mock tests for examination

,

Exam

,

How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

,

study material

,

ppt

,

Extra Questions

,

practice quizzes

,

How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

,

pdf

,

shortcuts and tricks

,

Free

,

Objective type Questions

,

Important questions

,

How to Solve Exponential Equations using Logarithms | General Aptitude for GATE - Mechanical Engineering

,

Sample Paper

;