Please help me solve the following. Thanks,
Write m^p = q using logarithms.
Solve (1/e)^2x = 14
by definition m^p = q is the same as
logm q = p
for your second question
(1/e)^2x = 14
e^(-2x) = 14
-2x = ln 14
x = -1/2 ln 14
Are you not familiar with the basic properties of logs and exponents?
These type of questions require you to know those.
To write m^p = q using logarithms, you can use the logarithmic property of exponents, which states that log(base m)(x^p) = p * log(base m)(x).
Using this property, you can rewrite the given equation: m^p = q
Take the logarithm of both sides with base m:
log(base m)(m^p) = log(base m)(q)
Applying the exponent property of logarithms on the left side:
p * log(base m)(m) = log(base m)(q)
Now, as log(base m)(m) = 1, the equation simplifies to:
p = log(base m)(q)
So, the equation m^p = q is equivalent to p = log(base m)(q).
Now let's move on to the next problem:
To solve the equation (1/e)^(2x) = 14, we can use logarithms to isolate the variable x.
Take the natural logarithm (ln) of both sides:
ln[(1/e)^(2x)] = ln(14)
Applying the logarithmic property of exponents on the left side:
2x * ln(1/e) = ln(14)
Now, we need to simplify the term ln(1/e).
Using the property ln(1/x) = -ln(x), we can rewrite it as:
-2x * ln(e) = ln(14)
Since ln(e) = 1, the equation becomes:
-2x = ln(14)
To isolate x, divide both sides by -2:
x = ln(14) / -2
So the solution to the equation (1/e)^(2x) = 14 is x = ln(14) / -2.