Exponetial Functions
In the following exponential relations, solve for x using logarithms. Round your answers to 4 decimal places.
a)
5^3x=65
b)
2^2x+3=80
Joseph/Matt/Mark/Chris -- please use the same name for your posts.
Okay,but can i get assistance?
To solve the exponential equations using logarithms, follow these steps:
Step 1: Take the logarithm of both sides of the equation. The choice of logarithm base is not important, but usually, logarithms with bases 10 (log) or e (ln) are used. In these examples, we will use the natural logarithm (ln).
Step 2: Apply logarithmic properties to simplify the equation.
Step 3: Solve for x by isolating it.
Now let's solve each of the given equations using logarithms:
a) 5^(3x) = 65
Step 1: Take the natural logarithm of both sides:
ln[5^(3x)] = ln(65)
Step 2: Apply the exponent rule for logarithms:
3x * ln(5) = ln(65)
Step 3: Solve for x by isolating it:
3x = ln(65) / ln(5)
x = (ln(65) / ln(5)) / 3
Calculate the right side using a calculator, then divide by 3 to get the approximate value of x.
b) 2^(2x+3) = 80
Step 1: Take the natural logarithm of both sides:
ln[2^(2x+3)] = ln(80)
Step 2: Apply the exponent rule for logarithms:
(2x+3) * ln(2) = ln(80)
Step 3: Solve for x by isolating it:
2x+3 = ln(80) / ln(2)
x = (ln(80) / ln(2)) - (3/2)
Calculate the right side using a calculator, then subtract 3/2 to get the approximate value of x, rounded to 4 decimal places.