algebra 2 problem

Question

The exponential equation has to be solved

5^x=2 

 

I'm trying to find the exact answer and solution

Answer 1

Any time you need to solve an equation with the unknown in the exponent, you are likely to need to use logarithms.  That gets the unknown out of the exponent.

 

First take the log of both sides.  You can use any base for the logarithm, so let's use base 10:

 

5^x=2

log (5^x) = log 2

 

Use the rule for exponents:

x log 5 = log 2

 

Divide both sides by log 5

x = log 2 / log 5

 

That's the exact answer.

 

 

Answer 2

Hello Lewis,

 

The answer is simple. First, take a Log base 5 of both sides:

 

Log₅(5^x)=Log₅(2)

x= Log₅(2)

 

Next, apply a transform using natural logarithm:

 

x= Ln(2)/Ln(5)

 

where Ln is a natural logarithm. It could also be written using decimal logarithm:

 

x= Log(2)/Log(5)

 

Just in case, the rounded value of x= 0.430676558

 

Hope this will help.