Solving Exponential Equation with Log

Steps:

• Take the logarithm of both sides.
• Use ln because we have a base of e.
• Then solve for the variable x.

Real-World Problem:

• How do we figure out growth rates? A country doesn't intend to grow at 8.56% per year. You look at the GDP one year and the GDP the next, and take the logarithm to find the implicit growth rate.

My two favorite interpretations of the natural logarithm (ln(x)), i.e. the natural log of 1.5:

• Assuming 100% growth, how long do you need to grow to get to 1.5? (.405, less than half the time period)
• Assuming 1 unit of time, how fast do you need to grow to get to 1.5? (40.5% per year, continuously compounded)

Logarithms are how we figure out how fast we're growing.

SAT Prep:

(e+1)k=100

• Take the common logarithm of both sides and solve for k:
• (e+1)k=100
• log(e+1)k=log100
• klog(e+1)=2
• k=2log(e+1)
• k=2log(2.7183+1)≈2log3.7183≈21.3133≈3.51