0. Functions

Solve the exponential equation.

$7^{2x^2-8}=1$

Rule of 70 Bankers use the Rule of 70, which says that if an account increases at a fixed rate of p%/yr, its doubling time is approximately 70/p. Use linear approximation to explain why and when this is true.

5. e^(2t)-3e^t = 0

Solve the exponential equation.

$900=10^{x+17}$

In Exercises 1–4, solve for t.

1. a. e^(-0.3t) = 27

Solve the exponential equation.

$100^{x}=10^{x+17}$

Compounded inflation The U.S. government reports the rate of inflation (as measured by the consumer index) both monthly and annually. Suppose for a particular month, the monthly rate of inflation is reported as 0.8%. Assuming this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.

Determine whether the following statements are true and give an explanation or counterexample.

(4x+1)^{ln x} = x^ln(4x+1)

Solve the logarithmic equation.

$\(\log\)_7\(\left\)(6x+13\(\right\))=2$

Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.

b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

In Exercises 1–4, solve for t.

2. b. e^(kt) = 10

Solve the exponential equation.

$81^{x+1}=27^{x+5}$

In Exercises 1–4, solve for t.

e^(sqrt(t)) = x^2

Solving equations Solve the following equations.

3(ˣ³⁻⁴) = 15

Solving equations Solve the following equations.

log₁₀ x= 3