How Do I Solve a Logarithmic Equation with Different Bases and Variables?

  • MHB
  • Thread starter Monoxdifly
  • Start date
  • Tags
    Logarithmic
In summary, the conversation was about solving the equation log_2(x+2)+log_{(x-2)}4=3. The speaker admitted to not knowing how to solve it due to the terms x+2 and x-2 being different. The friend mentioned that if they were the same, he could solve it easily. The speaker then showed their progress in solving the equation and asked for further guidance. The conversation ended with the consensus that no x value satisfies the equation.
  • #1
Monoxdifly
MHB
284
0
A friend asked me how to solve this question:
\(\displaystyle log_2(x+2)+log_{(x-2)}4=3\)
I said I had no idea because one is x + 2 and the other one is x - 2. If both are x + 2 or x - 2, I can do it. He said that if that's the case, even at his level he could solve it. This is what I've done so far regarding the question.
\(\displaystyle log_2(x+2)+log_{(x-2)}4=3\)
\(\displaystyle \frac{log(x+2)}{log2}+\frac{log4}{log(x-2)}=3\)
\(\displaystyle \frac{log(x+2)log(x-2)+log4log2}{log2log(x-2)}=3\)
\(\displaystyle log(x+2)log(x-2)+2log^22=3log2log(x-2)\)
\(\displaystyle log(x+2)log(x-2)-3log2log(x-2)=-2log^22\)
\(\displaystyle log(x-2)(log(x+2)-3log2)=-2log^22\)
What should I do from here? Or did I make some mistakes?
 
Mathematics news on Phys.org
  • #2
Beer hangover induced idea follows.
20200915_141613.jpg
 
  • #3
So, no x fulfills the equation, right?
 
  • #4
Monoxdifly said:
So, no x fulfills the equation, right?

That's what the graph says ...
 
  • #5
Okay, thanks guys.
 

FAQ: How Do I Solve a Logarithmic Equation with Different Bases and Variables?

What is a logarithmic equation?

A logarithmic equation is an equation in which the variable appears in the exponent of a logarithm. It is used to solve problems involving exponential growth and decay.

How do I solve a logarithmic equation?

To solve a logarithmic equation, you can use the properties of logarithms to rewrite the equation in a simpler form. Then, you can use algebraic methods to solve for the variable.

What are the properties of logarithms?

The properties of logarithms include the product rule, quotient rule, power rule, and change of base rule. These rules can be used to manipulate logarithmic equations and make them easier to solve.

Can a logarithmic equation have multiple solutions?

Yes, a logarithmic equation can have multiple solutions. This is because logarithms are not one-to-one functions, meaning that multiple inputs can result in the same output.

How are logarithmic equations used in real life?

Logarithmic equations are used in many real-life situations, such as calculating the pH level of a solution, measuring the intensity of earthquakes, and modeling population growth. They are also used in finance and economics to calculate compound interest and inflation rates.

Similar threads

MHB How to Solve for x in a Logarithmic Equation?
Replies
8
Views
3K
MHB Solving for an exponential equation using logarithms 16^{x}-5(4)^{x}-6=0
Replies
3
Views
2K
MHB Find Common Difference of A.P. Given G.P. & Logarithms
Replies
6
Views
2K
Writing a logarithm in a form not involving logarithms
Replies
4
Views
1K
MHB Why Are My Logarithmic Equation Solutions Extraneous?
Replies
1
Views
1K
Converting Logarithmic Equations to Base 2
Replies
2
Views
2K
MHB Change of bases with log tables
Replies
3
Views
2K
I How Can We Simplify Evaluating the Improper Integral of Logarithmic Functions?
Replies
10
Views
2K
MHB Finding x in Logarithmic Equation
Replies
1
Views
2K
MHB A logarithm formula involving the mascheroni constant
Replies
6
Views
2K

Hot Threads

Recent Insights

Back
Top