Master solving exponential and logarithmic equations using various algebraic techniques, including same-base method, logarithms, and change of base.
Solve the exponential equation!
Equation:
Solve the logarithmic equation!
Equation:
There are two main methods: (1) Same base method - if bases are equal, exponents are equal. (2) Logarithm method - take logarithm of both sides.
Solve: 2ˣ = 16
Step 1: Write 16 as a power of 2: 16 = 2⁴
Step 2: 2ˣ = 2⁴
Step 3: Since bases are equal, exponents are equal: x = 4
Answer: x = 4
Solve: 3ˣ = 7
Step 1: Take log of both sides: log(3ˣ) = log(7)
Step 2: Apply power rule: x·log(3) = log(7)
Step 3: Solve for x: x = log(7) / log(3) ≈ 1.771
Answer: x ≈ 1.771