-6.1.12 solve for x 3^{2x}=105

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  • Thread starter karush
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In summary, the conversation discusses how to solve the equation 3^(2x) = 105 and different ways to simplify the solution, including using logarithms and changing the base. The final suggestion is to use ln(105)/2ln(3) as the simplest solution, although there is some disagreement over whether it is truly simpler or not.
  • #1

karush

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solve for $\ x \quad 3^{2x}=105$
rewrite $x\ln3^2=x\ln{9}=\ln{105}$
hence $x=\dfrac{\ln{105}}{\ln{9}}$

can this be reduced more except for decimal
 
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  • #2
I would have done this a little differently:
2x ln(3)= ln(105)

x= ln(105)/2ln(3).

Of course, since 9= 3^2, ln(9)= 2ln(3) so that is the same as your answer. Whether it is "simpler" or not is a matter of personal choice.
 
  • #3
Why not just $\displaystyle x = \frac{1}{2}\log_3{\left( 105 \right) }$?
 
  • #4
change of base then..
 
  • #5
karush said:
change of base then..

I'm asking, why are you even bothering to change the base at all? The base of your exponential equation is 3, surely the best base for your logarithm would therefore also be 3...
 
  • #6
But my calculator doesn't HAVE a "logarithm base 3" key!
 
  • #7
Country Boy said:
But my calculator doesn't HAVE a "logarithm base 3" key!

And why are you bothering to use a calculator? Surely they would want an exact answer...
 

1. What is the value of x in the equation 3^(2x)=105?

The value of x in the equation 3^(2x)=105 is approximately 2.24.

2. How do I solve for x in the equation 3^(2x)=105?

To solve for x in the equation 3^(2x)=105, you can use logarithms. Take the logarithm of both sides of the equation, then use the power rule to bring down the exponent. Finally, solve for x using basic algebra.

3. Can I use a calculator to solve for x in the equation 3^(2x)=105?

Yes, you can use a calculator to solve for x in the equation 3^(2x)=105. You can use the logarithm function on your calculator to find the value of x.

4. Is there more than one solution to the equation 3^(2x)=105?

Yes, there is more than one solution to the equation 3^(2x)=105. In fact, there are two solutions: x = 2.24 and x = -2.24.

5. Can I use a different base for the logarithm when solving for x in the equation 3^(2x)=105?

Yes, you can use a different base for the logarithm when solving for x in the equation 3^(2x)=105. However, it is recommended to use the natural logarithm (ln) or base 10 logarithm (log) for simplicity.

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