Fractions and Decimals

Discrete Mathematics

Differential Equations

Diagram & Picture

Numbers & Operations

Data & Statistics

Natural Numbers

Ratios & Proportions

Order of Operations

Exponents & Radicals

Complex Numbers

Congruence & Similarity

Pythagorean Theorem

Transformations

Data Collection & Representation

Measures of Central Tendency

Measures of Spread

Statistical Inference

Non-Right Triangle

Unit Circle & Radians

Identities & Equations

Limits & Continuity

Sequences & Series

Solved on Nov 01, 2023

Solve the equation $(e^{3})^{2x-8} = e^{-4x+5}$

STEP 1

Assumptions1. The base of the exponential functions is $e$ , the natural logarithm base. . The equation is $\left(e^{3}\right)^{ x-8}=e^{-4 x+5}$

STEP 2

First, we simplify the equation. The power of a power rule states that $(a^m)^n = a^{mn}$ . So, we can simplify $\left(e^{}\right)^{2 x-8}$ to $e^{(2x-8)}$ .
$\left(e^{}\right)^{2 x-8} = e^{(2x-8)}$

STEP 3

Now, we simplify the expression $3(2x-8)$ to $6x-24$ .
$e^{3(2x-8)} = e^{6x-24}$

STEP 4

Now we have the equation $e^{6x-24}=e^{-4 x+}$ .

STEP 5

Since the bases are the same, we can equate the exponents. This gives us the equation $x-24=-4x+5$ .
$x-24=-4x+5$

STEP 6

Now, we solve the equation for $x$ . First, we add $4x$ to both sides of the equation to get $10x-24=5$ .
$10x-24=5$

STEP 7

Next, we add24 to both sides of the equation to get $10x=29$ .
$10x=29$

STEP 8

Finally, we divide both sides of the equation by10 to solve for $x$ . $x=\frac{29}{10}$ The solution to the equation $\left(e^{3}\right)^{2 x-8}=e^{-4 x+5}$ is $x=\frac{29}{10}$ .

Was this helpful?

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Contact Influencer program Policy Terms