In this lesson:
By the end of this lesson, you should be able to:
You know from inverse function theory: if and are inverses, then:
Applied here: and — so each undoes the other.
Order 1: Apply exponential, then logarithm:
Order 2: Apply logarithm, then exponential:
Both hold because and are inverse functions
Examples:
The exponent and the logarithm are the same base — they cancel
Simplify :
The exponent is — same base as , so they cancel:
Simplify : same reasoning → result is
Check: does the base of the exponent match the base of the log?
The logarithm returns the exponent — even when the exponent is negative
Same base → they cancel and return the inner value
Simplify:
For each: identify the bases — do they match? What does that tell you?
Answers: · ·
When the unknown is in the exponent — apply to both sides:
The logarithm "brings down" the exponent, making the subject
Solve:
By inspection: , so
Verify using logarithms:
Since , the what-exponent question confirms ✓
Clean answers like this confirm the logarithm method works
No integer power of 3 gives exactly 20 — can't solve by inspection.
Apply log:
Change-of-base:
The unknown is inside the logarithm — apply to both sides:
Exponentiation cancels the logarithm; steps out
The unknown is inside the log — exponentiate both sides base 2:
First exponentiate to escape the log, then solve the remaining linear equation
Step 1: Apply to both sides:
Step 2: Change-of-base — choose or :
Complete Step 2 with a calculator before the next slide
Solve each equation completely:
Label your move for each: "apply log" or "apply exponentiation"
because (clean answer)
✓ and ✓ → apply → ✓ → apply →
Bases must match for cancellation Argument — check after solving
You can now solve both exponential and logarithmic equations.
Coming up:
The inverse relationship is the thread connecting all of logarithm theory