📝 Chapter 4: Solving Exp. Equations & Applications

Solve: \( 5^{x-2} = 125 \).

Solve: \( 2^{3x+1} = 32^{x-2} \).

Solve: \( 4^{x+1} = 8^{x} \).

Population grows at 3% per year. Starting at 1500, find the population after 10 years (round to nearest whole).

A radioactive substance has half-life 10 days. From 240 mg, find the mass after 30 days.

A bacterial colony has 100 cells at 8 a.m. and 800 cells at noon. Determine the doubling time. Show all reasoning.

Solve: \( 9^{x} \cdot 27^{x-1} = \dfrac{1}{3} \). (Hint: write all terms as powers of 3.)

A 10-gram sample of an isotope is reduced to 1.25 grams in 24 hours. Find the half-life. Show all reasoning.

Explain how to recognise when an exponential equation can be solved by the same-base method. What if it can't?

Compare the form \( A_0(1 + r)^t \) with the form \( A_0 \cdot b^{t/T} \). Translate the model "100 g, doubles every 5 years" into both forms.

Why does compounding more frequently (e.g. monthly vs. annually) yield a larger amount? Explain using \$1000 at 6% over 1 year as your numerical example.

Translate the phrase "decreases by 8% per year" into a multiplicative factor and a base. Then write the function for an initial value of 5000.

\$5000 is invested at 6% compounded monthly. (a) Write the formula for the value \( A(t) \) after \( t \) years. (b) Find the value after 8 years. (c) Compare with simple interest at 6%.

A car worth \$30{,}000 depreciates 12% per year. (a) Write \( V(t) \). (b) Find \( V(5) \). (c) After how many years is the car first worth less than \$10{,}000? (Use a table; round up.)

Carbon-14 has a half-life of 5730 years. A bone fragment shows 25% of the original C-14. Estimate the age of the fragment. Show all reasoning.