solve the following exponential equation. exact answers only
1-8x 3x
¨i =e
*I know that I did not post this correctly but I do not know how to put the 1-8x above the pie symbol and 3x above the e.
Normally that would be rendered as
pi^(1-8x) = e^(3x)
so, take ln of both sides:
(1-8x) ln(pi) = 3x
ln(pi) - 8ln(pi) x = 3x
x(3+8ln(pi)) = ln(pi)
x = ln π/(3 + 8 ln π)
Thanks for telling me how to actually write it here.
No problem! I understand what you're looking for. To solve the exponential equation 1-8x^(3x) = e, you need to use logarithms to eliminate the exponent and solve for x.
Step 1: Start by taking the natural logarithm (ln) of both sides of the equation. This will remove the exponential function:
ln(1-8x^(3x)) = ln(e)
Step 2: Simplify the equation using logarithmic properties. The natural logarithm of e is simply 1:
ln(1-8x^(3x)) = 1
Step 3: Now, we need to solve for x. To do so, we have to isolate x. First, we can exponentiate both sides of the equation using the base e:
e^(ln(1-8x^(3x))) = e^1
Step 4: The exponential function and the natural logarithm are inverse functions, so they cancel each other out:
1 - 8x^(3x) = e
Step 5: Rearrange the equation to isolate the term with the exponent:
8x^(3x) = 1 - e
Step 6: To solve this equation, we need to rewrite it in a form where both sides are raised to the same power. Taking the natural logarithm of both sides can help us achieve this:
ln(8x^(3x)) = ln(1 - e)
Step 7: Simplify the expression:
(3x)ln(8x) = ln(1 - e)
Step 8: Divide both sides of the equation by ln(8x) to isolate the variable:
3x = ln(1 - e) / ln(8x)
Step 9: Finally, divide both sides by 3 to solve for x:
x = (ln(1 - e) / ln(8x)) / 3
Note: It is not possible to find an exact numerical solution for this equation without knowing the specific values of e and x. However, this is the exact algebraic solution to the exponential equation.