An exponential function "f" is defined by f(x)=7(1.5)^x-6. if f(k)=91, then what is the value of k?
you are solving
7(1.5)^(k-6) = 91
1.5^(k-6) = 13
take log of both sides
log [1.5^(k-6)] = log 13
(k-6)[log 1.5] = log 13
k-6 = log13/log1.5 = 6.3259
k = 12.3259
check
7(1.5^6.3259)
= 7(12.99977) = 90.9984 (not bad)
log1/7(x^2=x)-log1/7(x^2-x)=-1
To find the value of k, we need to use the information given in the problem and solve for k in the equation f(k) = 91.
The exponential function f(x) is defined as f(x) = 7(1.5)^(x-6). We are given that f(k) = 91.
Substituting these values into the equation, we have:
7(1.5)^(k-6) = 91
To solve for k, we will follow these steps:
Step 1: Divide both sides of the equation by 7 to isolate the exponential term:
(1.5)^(k-6) = 13
Step 2: Take the logarithm of both sides of the equation. Since the base of the exponential term is 1.5, we can use either the natural logarithm (ln) or the logarithm base 1.5 (log₁.₅). In this case, we will use the natural logarithm (ln) for simplicity:
ln(1.5)^(k-6) = ln(13)
Step 3: Use the logarithmic property to bring down the exponent:
(k-6) * ln(1.5) = ln(13)
Step 4: Divide both sides of the equation by ln(1.5) to solve for k:
k-6 = ln(13) / ln(1.5)
Step 5: Add 6 to both sides of the equation to solve for k:
k = ln(13) / ln(1.5) + 6
Using a calculator, we can evaluate the right side of the equation to find the value of k.