Exam 5: Exponential and Logarithmic Functions and Equations

Solve the problem. -Suppose that $6000 is invested at an interest rate of 5.2% per year, compounded continuously. What is the balance after 2 years?Round to the nearest cent.

Solve the problem. -How long will it take for $3000 to grow to $38,800 at an interest rate of 6.2% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

Solve the equation. - $\operatorname { log } ( x + 2 ) = \log ( 5 x + 1 )$

Use the properties of logarithms to evaluate the expression without the use of a calculator. - $\log _ { 3 } 3 ^ { - 4 }$

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $3 ^{( x - 1 )} = 16$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 7 } \sqrt [ 5 ] { \frac { x ^ { 4 } y } { 49 } }$

Write the exponential equation as an equation involving a common logarithm or a natural logarithm. - $\mathrm { e } ^ { \mathrm { x } } = 23$

Write the logarithmic equation as an exponential equation. - $\ln 1 = 0$

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $3 ^ { x + 6 } = 5$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log \left( \frac { x } { 10 } \right)$

Solve the equation. - $\log _ { 2 } ( x - 5 ) + \log _ { 2 } ( x - 11 ) = 4$

Solve the problem. -An original investment of $7000 earns 7% interest compounded continuously. What will the investment be worth in 3 years? Round to the nearest cent.

Solve the equation. - $\left( e ^ { x } \right) ^ { x } \cdot e ^ { 32 } = e ^ { 12 x }$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\frac { 1 } { 3 } \left( \log _ { 4 } x + \log _ { 4 } y \right)$

Write the logarithmic equation as an exponential equation. - $\log _ { 2 } x = 3$

Solve the equation. - $3 ^ { 6 - 3 x } = \frac { 1 } { 27 }$

Solve the problem. -Find the amount in a savings account at the end of 4 years if the amount originally deposited is $9000 and the interest rate is 5.5% compounded monthly. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ where: $A =$ final amount $\mathrm { P } = \$ 9000$ (the initial deposit) $\mathrm { r } = 5.5 \% = 0.055$ (the annual rate of interest) $\mathrm { n } = 12$ (the number of times interest is compounded each year) $t = 4$ (the duration of the deposit in vears)

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $5 \ln ( x - 11 ) - 8 \ln x$

Solve the problem. -There are currently 57 million cars in a certain country, increasing exponentially by 0.8% annually. How many years will it take for this country to have 64 million cars? Round to the nearest year.

Solve the equation. - $2 ^ { 1 + 2 x } = 8$