Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Consider the equation

0.5*e^(4z)=13". "
Solve the equation for
z. Express the solution as a logarithm in base
e.

z=
Approximate the value of
z. Round your answer to the nearest thousandth.

z~~

Consider the equation $\newline$ $0.5 \cdot e^{4 z}=13 \text {. }$ $\newline$ Solve the equation for $z$ . Express the solution as a logarithm in basee. $\newline$ $z=$ $\newline$ Approximate the value of $z$ . Round your answer to the nearest thousandth. $\newline$ $z \approx$

View step-by-step help

View step-by-step help

Change of base formula

Full solution

Q. Consider the equation

\newline

0.5 \cdot e^{4 z}=13 \text {. }

\newline

Solve the equation for

z

. Express the solution as a logarithm in basee.

\newline

z=

\newline

Approximate the value of

z

. Round your answer to the nearest thousandth.

\newline

z \approx

Isolate exponential term: Isolate the exponential term $e^{4z}$ . $\newline$ To isolate $e^{4z}$ , we need to divide both sides of the equation by $0.5$ . $\newline$ $0.5 \cdot e^{4z} = 13$ $\newline$ $e^{4z} = \frac{13}{0.5}$ $\newline$ $e^{4z} = 26$
Take natural logarithm: Take the natural logarithm of both sides. $\newline$ To solve for $z$ , we take the natural logarithm ( $\ln$ ) of both sides of the equation because the natural logarithm is the inverse function of the exponential function with base $e$ . $\newline$ $\ln(e^{4z}) = \ln(26)$
Apply logarithm property: Apply the property of logarithms that $\ln(e^x) = x$ . $\newline$ Using this property, we can simplify the left side of the equation. $\newline$ $4z = \ln(26)$
Solve for z: Solve for z. $\newline$ To solve for z, we divide both sides of the equation by $4$ . $\newline$ $z = \frac{\ln(26)}{4}$
Approximate z: Approximate the value of z. $\newline$ Using a calculator, we can find the approximate value of $\ln(26)$ and then divide by $4$ . $\newline$ $z \approx \frac{\ln(26)}{4}$ $\newline$ $z \approx \frac{3.2581}{4}$ $\newline$ $z \approx 0.8145$

More problems from Change of base formula

Question

\newline

\frac{5}{6} \div \frac{9}{4}=

Posted 3 months ago

Question

\newline

\frac{8}{3} \div \frac{5}{3}=

Posted 3 months ago

Question

\newline

\frac{5}{4} \div \frac{3}{5}=

Posted 3 months ago

Question

\newline

\frac{3}{4} \div \frac{3}{4}=

Posted 3 months ago

Question

Write the expression in exponential form.

\newline

6 \times 6 \times 6 \times 6 \times 6=

Posted 3 months ago

Question

Write the expression in exponential form.

\newline

9 \times 9 \times 9 \times 9 \times 9 \times 9 \times 9=

Posted 3 months ago

Question

The value of Vishal's car is depreciating exponentially.

\newline

The relationship between

V

, the value of his car, in dollars, and

t

, the elapsed time, in years, since he purchased the car is modeled by the following equation.

\newline

V=22,500 \cdot 10^{-\frac{t}{12}}

\newline

How many years after purchase will Vishal's car be worth

\$ 10,000

\newline

Give an exact answer expressed as a base

-10

\newline

Posted 3 months ago

Question

A scientist measures the initial amount of Carbon

-14

in a substance to be

25

\newline

The relationship between

A

, the amount of Carbon

-14

remaining in that substance, in grams, and

t

, the elapsed time, in years, since the initial measurement is modeled by the following equation.

\newline

A=25 e^{-0.00012 t}

\newline

In how many years will the substance contain exactly

20

(\mathrm{g})

-14

\newline

Give an exact answer expressed as a natural logarithm.

\newline

Posted 3 months ago

Question

Which of the following is equivalent to

\log _{7}(a) \cdot \log _{b}(7)

\newline

1

\newline

\log (7)

\newline

\log (7 a)

\newline

\log _{a}(b)

\newline

\log _{b}(a)

Posted 3 months ago

Question

Which of the following is equivalent to

\log (a) \cdot \log _{a}(5)

\newline

1

\newline

\log (a)

\newline

\log (5)

\newline

\log (5 a)

\newline

\log _{a}(5 a)

Posted 3 months ago