Find x:
2(5)^x=3^(x+1)
I'm not sure what to do with the 2. I think I need to write the equation as log5^2x=log3^(x+1), but I don't know.
Thanks
If it is supposed to be
2 * 5^x = 3^(x+1)
then just take logs as usual:
log(2) + xlog(5) = (x+1)log(3)
log2 + xlog5 = xlog3 + log3
x(log5-log3) = log3-log2
x = (log3-log2)/(log5-log3)
or, if you prefer,
x = log(3/2)/log(5/3)
Thanks. That really helped!
To solve the equation 2(5)^x = 3^(x+1) for x, we can first simplify the equation before taking logarithms. Let's work on that:
Start with the equation:
2(5)^x = 3^(x+1)
First, we can rewrite 5 as the prime factorization of 3^x:
2(3^x)^x = 3^(x+1)
Next, apply the power of a power rule:
2(3^x • 3^x) = 3^(x+1)
Multiply the terms on the left side:
2(3^2x) = 3^(x+1)
Now, we have two options to proceed further:
Option 1: Using logarithms:
To eliminate the exponents, we can take the logarithm of both sides of the equation. Here, we can use either the logarithm base 3 or base 10. Let's use the natural logarithm (base e) since it is widely used:
ln[2(3^2x)] = ln[3^(x+1)]
Apply the logarithm rules:
ln(2) + ln(3^2x) = (x+1)ln(3)
Use the power rule of logarithms to simplify the equation further:
ln(2) + 2xln(3) = xln(3) + ln(3)
Combine like terms:
2xln(3) - xln(3) = ln(3) - ln(2)
Factor out x:
x(2ln(3) - ln(3)) = ln(3) - ln(2)
Simplify:
x ln(3) = ln(3) - ln(2)
Finally, divide both sides of the equation by ln(3):
x = (ln(3) - ln(2)) / ln(3)
Option 2: Manipulating the equation further:
From the equation we obtained earlier:
2(3^2x) = 3^(x+1)
We can divide both sides by 3^x to simplify the equation:
2(3^x) = 3 • 3^x
Now, we divide both sides by 3:
2 = 3^x
Take the logarithm of both sides:
log₂(2) = log₂(3^x)
Using the power rule of logarithms:
log₂(2) = x log₂(3)
Finally, divide both sides by log₂(3):
x = log₂(2) / log₂(3)
To find the approximate value of x, you can evaluate the expression using a calculator.