Solve the following exponential equation. Express the solution set in terms of natural logarithms or common loganthms. Then, us

The solution set expressed in terms of logarithms is , In 3.
Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression
Now use a calculator to obtain a decimal approximation for the solution.
The solution set is

Ask by Wagner Bates. in Saudi Arabia

Mar 09,2025

Question

Solve the following exponential equation. Express the solution set in terms of natural logarithms or common loganthms. Then, us

The solution set expressed in terms of logarithms is , In 3.
Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression
Now use a calculator to obtain a decimal approximation for the solution.
The solution set is

Ask by Wagner Bates. in Saudi Arabia

Mar 09,2025

UpStudy AI · Accepted Answer

We start with the given equation  
$$
e^{2x} - 4e^x + 3 = 0.
$$
Introduce the substitution  
$$
u = e^x.
$$
Then the equation becomes  
$$
u^2 - 4u + 3 = 0.
$$

Factor the quadratic:  
$$
(u-1)(u-3)=0.
$$
This gives the solutions  
$$
u = 1 \quad \text{or} \quad u = 3.
$$

Recall that $$ u = e^x $$. Thus, for $$ u=1 $$ we have  
$$
e^x = 1 \quad \Longrightarrow \quad x = \ln(1) = 0.
$$
For $$ u=3 $$ we have  
$$
e^x = 3 \quad \Longrightarrow \quad x = \ln(3).
$$

So the solution set expressed in terms of logarithms is  
$$
\{0,\, \ln(3)\}.
$$

Using a calculator, $$ \ln(3) $$ is approximately  
$$
\ln(3) \approx 1.10.
$$

Thus, the solution set with decimal approximations is  
$$
\{0,\, 1.10\}.
$$
The solution set is $$ \{0, \ln(3)\} $$, which is approximately $$ \{0, 1.10\} $$.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions