How do you solve for the exponent in an exponential function?

[wunnymush13]

I need to solve for the exponent "x" in the equation 7 • 1.0111^x = 10 ... problem is I'm not positive on how to? Can I put the 7 on the other side of the equation and make it 10-7 = 1.0111^x? I don't know how to follow through on this entirely..

 

[soroban]

Hello, wunnymush13!

\(\displaystyle \text{Solve for }x:\;\;7\cdot1.0111^x \:=\: 10\)

If you think you can eliminate the 7 by subtracting 7 from both sides,
. . you need more help than we can offer.


We have: .. . \(\displaystyle 7\cdot 1.0111^x \:=\:10\)

Divide by 7: .. ..\(\displaystyle 1.0111^x \:=\:\frac{10}{7}\)

Take logs: .\(\displaystyle \ln\left(1.0111^x\right) \:=\:\ln\left(\frac{10}{7}\right)\)

. . . . . . . . \(\displaystyle x\!\cdot\!\ln(1.0111) \:=\:\ln\left(\frac{10}{7}\right)\)

. . . . . . . . . . . . . . . . \(\displaystyle x \:=\:\dfrac{\ln\left(\frac{10}{7}\right)}{\ln(1.0111)}\; \approx\;32.31 \)