Studdy AI Logo
Math  /  Algebra

Question11. The equation f(x)=7a2+x−bf(x)=7 a^{2+x}-b, has an xx - intercept equivalent to A. x=log⁡b−log⁡7log⁡a−2x=\frac{\log b-\log 7}{\log a}-2 B. x=7a2−bx=7 a^{2}-b C. x=y+b7a2x=\frac{y+b}{7 a^{2}} D. x=0x=0

Studdy Solution

STEP 1

1. The function f(x)=7a2+x−b f(x) = 7a^{2+x} - b is given.
2. An x x -intercept occurs where f(x)=0 f(x) = 0 .
3. We are solving for x x when the function equals zero.

STEP 2

1. Set the function equal to zero.
2. Solve the equation for x x .
3. Simplify the expression to match one of the given options.

STEP 3

Set the function equal to zero to find the x x -intercept:
7a2+x−b=0 7a^{2+x} - b = 0

STEP 4

Solve the equation for x x :
First, add b b to both sides:
7a2+x=b 7a^{2+x} = b
Next, divide both sides by 7:
a2+x=b7 a^{2+x} = \frac{b}{7}

STEP 5

Take the logarithm of both sides to solve for x x . We use the property of logarithms that allows us to bring the exponent down:
log⁡(a2+x)=log⁡(b7) \log(a^{2+x}) = \log\left(\frac{b}{7}\right)
This simplifies to:
(2+x)log⁡a=log⁡b−log⁡7 (2+x) \log a = \log b - \log 7

STEP 6

Solve for x x by isolating it:
2log⁡a+xlog⁡a=log⁡b−log⁡7 2 \log a + x \log a = \log b - \log 7
Subtract 2log⁡a 2 \log a from both sides:
xlog⁡a=log⁡b−log⁡7−2log⁡a x \log a = \log b - \log 7 - 2 \log a
Divide by log⁡a \log a to solve for x x :
x=log⁡b−log⁡7−2log⁡alog⁡a x = \frac{\log b - \log 7 - 2 \log a}{\log a}
This simplifies to:
x=log⁡b−log⁡7log⁡a−2 x = \frac{\log b - \log 7}{\log a} - 2

STEP 7

Compare the simplified expression for x x with the given options. The expression matches:
A. x=log⁡b−log⁡7log⁡a−2 x = \frac{\log b - \log 7}{\log a} - 2
The correct answer is A \boxed{A} .

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord