Solve the exponential equation. Round the answer to four decimal places, if necessary.
A) x = 0.3869
B) x = 3
C) x = 2.585
D) x = 12
E) x = 0.453

Question

Solve the exponential equation. Round the answer to four decimal places, if necessary. 
A) x = 0.3869 
B) x = 3 
C) x = 2.585 
D) x = 12 
E) x = 0.453

Quizplus · Accepted Answer

We can rewrite the exponential equation as:

$3^{x-1} = 10 - 3^x$

Now, we can solve for x numerically using a graphing calculator or computer software. One way to do this is to plot the two functions $y=3^{x-1}$ and $y=10-3^x$ and look for their intersection point(s).

Alternatively, we can use algebraic techniques to rearrange the equation into a more convenient form. For example, we can rewrite it as:

$3^{x-1} + 3^x - 10 = 0$

Let $y = 3^{x-1}$. Then, the equation becomes:

$y + 3y - 10 = 0$

Solving for y, we get:

$y = \frac{10}{4} = 2.5$

Substituting back, we have:

$3^{x-1} = 2.5$

Taking the logarithm of both sides (with any base), we get:

$(x-1) \log 3 = \log 2.5$

Solving for x, we get:

$x = \frac{\log 2.5}{\log 3} + 1 \approx 2.5848$

Rounding to four decimal places, we get:

$x \approx 2.585$

Solve the Exponential Equation