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
5^3x = 65
3x log5 = log65
3x = log65/log5
x = log65/3log5
I assume you meant
2^(2x+3) = 80
(2x+3)log2 = log80
2x+3 = log80/log2
x = (log80/log2 - 3)/2
a)
The way you typed it ....
125x = 65
x = 65/125 = 13/25
if you meant: 5^(3x) = 65
take log of both sides, then use log rules
3x log5 = log65
x = log65/(3log5) = appr .865
b) again, the way you typed it...
4x + 3 = 80
4x = 77
x = 77/4
did you mean 2^(2x + 3) = 80 or did you mean
2^(2x) + 3 = 80 ?
Can you see how essential brackets have to be in this type of problem
To solve for x in exponential equations using logarithms, we can take the logarithm of both sides of the equation. There are two commonly used logarithms in mathematics: the natural logarithm (ln), which has base e, and the common logarithm (log), which has base 10. You can use either of these logarithms to solve the given equations. I will demonstrate both methods for you to choose from.
a)
Given: 5^(3x) = 65
Method 1: Using the natural logarithm (ln)
Taking the logarithm of both sides of the equation:
ln(5^(3x)) = ln(65)
Using the logarithmic property where ln(a^b) = b * ln(a):
3x * ln(5) = ln(65)
Divide both sides by ln(5) to isolate 3x:
3x = ln(65) / ln(5)
Now, divide both sides by 3 to solve for x:
x = (ln(65) / ln(5)) / 3
Now you can use a calculator to find the value of x. Round your answer to 4 decimal places.
Method 2: Using the common logarithm (log)
Taking the logarithm of both sides of the equation:
log(5^(3x)) = log(65)
Using the logarithmic property where log(a^b) = b * log(a):
3x * log(5) = log(65)
Divide both sides by log(5) to isolate 3x:
3x = log(65) / log(5)
Now, divide both sides by 3 to solve for x:
x = (log(65) / log(5)) / 3
Using a calculator, find the value of x, rounding your answer to 4 decimal places.
b)
Given: 2^(2x + 3) = 80
Again, we will use logarithms to solve for x. You can choose between the natural logarithm (ln) or the common logarithm (log).
Method 1: Using the natural logarithm (ln)
Taking the logarithm of both sides of the equation:
ln(2^(2x + 3)) = ln(80)
Using the logarithmic property where ln(a^b) = b * ln(a):
(2x + 3) * ln(2) = ln(80)
Now, divide both sides by ln(2) to isolate 2x + 3:
2x + 3 = ln(80) / ln(2)
Next, subtract 3 from both sides to solve for 2x:
2x = (ln(80) / ln(2)) - 3
Finally, divide both sides by 2 to find x:
x = [(ln(80) / ln(2)) - 3] / 2
Using a calculator, determine the value of x, rounding your answer to 4 decimal places.
Method 2: Using the common logarithm (log)
Taking the logarithm of both sides of the equation:
log(2^(2x + 3)) = log(80)
Using the logarithmic property where log(a^b) = b * log(a):
(2x + 3) * log(2) = log(80)
Now, divide both sides by log(2) to isolate 2x + 3:
2x + 3 = log(80) / log(2)
Next, subtract 3 from both sides to solve for 2x:
2x = (log(80) / log(2)) - 3
Finally, divide both sides by 2 to find x:
x = [(log(80) / log(2)) - 3] / 2
Again, using a calculator, calculate the value of x, rounding your answer to 4 decimal places.