Archive for the ‘logarithmic’ Category

Problem 2021

Sales of Osgood’s Nature Soap have increased by 15% annually since 2000. Let f(t) be the annual sales if Osgood’s Natural Soap in thousands of bars, t years after 2000. The formula for f is:

f(t) = 5e0.14t

Answer the following question.

In how many years will the annual sales reach 10 thousand bars?

 

Solution:-

 

In how many years will the annual sales reach 10 thousand bars?

4.95

 

 

Problem 2020

Solve the logarithmic equation.

380 + 20 log x = 40

 

Solution:-

 

The solution is x = 10-17

 

Problem 2019

Solve the logarithmic equation.

ln (x) + ln (x – 2) = ln 3

 

Solution:-

 

The solution is x = 3

 

Problem 2018

Solve the logarithmic equation.

log5x + log5(x+2) = log53

 

Solution:-

 

Rewrite the left side of the equation as one logarithm, using the properties of logarithms.

log5(x(x+2)) = log53

Next, exponentiate both sides of the equation.

x(x+2) = 3

Now, solve for x. Use the Distributive Property.

x2 + 2x = 3

Set the equation equal to zero.

x2 + 2x -3 = 0

Solve the resulting quadratic equation. Use the Quadratic Formula.

x = \frac{-2\pm \sqrt{16}}{2}

Remember that the domain of logmx is all x > 0, so the answer must be greater than zero.

x = \frac{-2\pm \sqrt{16}}{2}

=1

 

Problem 2017

Solve the logarithmic equation.

210 + 10 log x = 30

 

Solution:-

 

210 + 10 log x = 30

10 log x = -180

log x = -18

Next exponential each side of the equation. Since the equation is in terms of the common logarithm, use base 10. The rule exponentiation states that if m = n, then 10m = 10n.

Log x = -18

10logx = 10-18

Finally, apply the inverse property 10log x = x.

10logx = 10-18

x = 10-18

Thus, the solution is x = 10-18.

 

Problem 1115

Solve the logarithmic equation. Be sure to reject any value of x that is not the domain of the original logarithmic expression.

6 In(8x) = 12

What is the exact solution?

What is the decimal approximation to the solution?

 

Solution:-

 

The solution set is {\frac{e^{2}}{8}}.

The decimal solution set is {0.92}.

 

 

Problem 1114

Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain if the original logarithmic expression. Give the exact answer.

{log_{2}}^{3x+3} = 4

 

Solution:-

 

The solution set is {\frac{13}{3}}.

 

 

Problem 1113

Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer.

{log_{4}}^{x}.

 

Solution:-

 

The solution set is {64}

 

 

Problem 877

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic, expressions.

log(1,000,000 y)

 

Solution:-

 

the logarithmic expression is

log(1,000,000 y) = 6 + log y

 

 

Problem 876

Use the properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions.

{log_{4}}^{13*11}

 

Solution:-

 

The logarithmic expression

{log_{4}}^{13*11} = {log_{4}}^{13} +  {log_{4}}^{11}