## 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) = 5e^{0.14t}

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

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## 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.

log_{5}^{x} + log_{5}^{(x+2)} = log_{5}^{3}

**Solution:-**

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

log_{5}^{(x(x+2))} = log_{5}^{3}

Next, exponentiate both sides of the equation.

x(x+2) = 3

Now, solve for x. Use the Distributive Property.

x^{2} + 2x = 3

Set the equation equal to zero.

x^{2} + 2x -3 = 0

Solve the resulting quadratic equation. Use the Quadratic Formula.

x =

Remember that the domain of log_{m}x is all x > 0, so the answer must be greater than zero.

x =

=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 10^{m} = 10^{n}.

Log x = -18

10^{logx} = 10^{-18}

Finally, apply the inverse property 10^{log x} = x.

10^{logx } = 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 .

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.

= 4

**Solution:-**

The solution set is .

## 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.

.

**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.

*Solution:-*

The logarithmic expression

=