Exam 20: Exponential and Logarithmic Functions

Solve for $x:\left(2^{x+3}\right)\left(3^{x+2}\right)=9\left(24^{x}\right)$

Solve for $x: e^{4 x-3}=2 e^{2 x+1}$

Graph the exponential function over the given domain of $x$ . $y=4-4 e^{-x}(x=0$ to 4 $)$

In a certain chemical reaction the concentration of a certain product molecule $C$ is given by $C=2\left(1-e^{-0.13 t}\right)$ moles/liter, where $t$ is the time the reaction has been running in seconds. Find $C$ after the reaction has run for eleven seconds.

Simplify: $10^{2 \log x}$

Solve for $x: 3^{x+1}+3^{2-x}=12$

Convert to exponential form: $\log _{5} x=3 y$

A person has $\$ 25,000$ is invested in an account that pays $6 \%$ interest per year, compounded monthly. What is the value of the account after ten years?

Solve for $x$ to three significant digits: $\ln (2 x+5)-\ln (6 x+1)=\ln (6)$

Solve for $x: \sqrt[x]{81}-12 \sqrt[2 x]{81}+27=0$

A certain company comes out with a new product and the number of investors in the company grows exponentially at a rate of $10 \%$ per week. After three weeks, there are 1,290 investors in the company. How many investors were there before the three weeks?

Solve for $x$ , to three significant digits: $2^{4 x}+2^{2 x}=132$

If the power input is $P_{1}$ and the power output is $P_{2}$ , the number of decibels gained or lost is $G=10 \log \frac{P_{2}}{P_{1}}$ (dB). If the amplifier gains $15.0 \mathrm{~dB}$ for an input of $90.0 \mathrm{~W}$ , find the power output.

Find the common logarithm of each number to four decimal places: (a) 0.5 (b) 12.75 (c) 291 (d) 0.0132

Convert to logarithmic form: $e^{\pi t}=v$

Solve for $x: \log _{x}\left(x^{2}-2 x+4\right)=2$

Solve for $x: 2^{x}+2^{3-x}=6$

Rewrite the following equation so that it contains no logarithms: $\log (x+y)+\log 2=\log \left(x^{2}-y^{2}\right)$

How many years does a person need to leave $\$ 8000$ in an account that pays $6.5 \%$ interest per year, compounded annually, to end up with $\$ 13,240$ ?

Simplify: $e^{-3 \ln x}$