SOLUTION: How do you solve the exponential equation in this problem? 3^x=243

Question 268217: How do you solve the exponential equation in this problem?
3^x=243

Found 2 solutions by jsmallt9, dabanfield:
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

So we are trying to find the power of 3 that is equal to 243. Well

So the answer is 5.

If the right side had been 242 or if we din't even think to see if 243 is a whole number power of 3, then we would use logarithms to solve an equation like this. Use a logarithm your calculator "knows", like base 10 or base e (ln):

Now we use a property of logarithms, , to move the coefficient out in front. (This property is the reason for using logarithms. It gives us a way to get the variable out of the exponent.):

Divide both sides by :

If you use your calculator to find both of these logarithms and then divide them, you will find that the answer is 5 (or a decimal number very, very close to 5. Remember not even your calculator know what log(243) or log(3) are exactly. It must use decimal approximations for them. And these decimal approximations will have round-off errors which may result in an answer like 5.00000001 or 4.9999999.)
Answer by dabanfield(803) (Show Source): You can put this solution on YOUR website!
How do you solve the exponential equation in this problem?
3^x=243

If we take the log (base 10) of both sides we have:
log (base 10) 3^x = log (base 10) 243
Using the law that says log a^b = b*log a we have:
x*log (base 10) 3 = log (base 10) 243
x = [log (base 10) 243]/[log (base 10) 3]
x = [log (base 10 (2.43*100)]/[log (base 10) 3]
Using the law that says log (a*b) = log a + log b:
x = [log (base 10) 2.43 + log (base 10) 100]/[log (base 10) 3]
Since the log (base 10) of 100 = 2 (remember 10^2 = 100 so the log (base 10) of 100 = 2):
x = [(log (base 10) 2.43) + 2]/[log (base 10) 3.
Look up the base 10 logs for 2.43 and 3 and finish the calculation.

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