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Chapter 5: Q. 13 (page 352)

Solve the equation 4^x-3 = 8^2x.

Short Answer

Expert verified

Value of $x i s - \frac{3}{2}$

Step by step solution

01

Step 1. Given Information:

Given Equation: 4^x-3 = 8^2x

We want to solve the given equation for value of x.

02

Step 2. Solution

Notice how bases 4 and 8 can be reduced to the same base, which is a = 2.

We will use the property ${(a^{m})}^{n} = a^{m \cdot n}$

role="math" localid="1647415498328" $4^{x - 3} = 8^{2 x} {(2^{2})}^{x - 3} = {(2^{3})}^{2 x} 2^{2 (x - 3)} = 2^{3 \cdot 2 x} 2^{2 x - 6} = 2^{6 x}$

Now that we have reduced the expressions to equal bases we will use the following property:

role="math" localid="1647415462256" $I f a^{x} = a^{y}, t h e n x = y (w h e r e a \neq 1 a n d a > 0) - - - (1)$

Using the property (1) we get

$2^{2 x - 6} = 2^{6 x} 2 x - 6 = 6 x 2 x - 6 x = 6 - 4 x = 6 x = - \frac{3}{2}$

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Most popular questions from this chapter

If $f (x) = a^{x}$ , show that $f (A + B) = f (A) . f (B)$ .

Begin with the graph of $y = e^{x}$ and use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

$f (x) = 7 - 3 e^{2 x}$

The domain of a one-to-one functiong is $(- \infty, 0)$ , and its range is $[0, \infty)$ . State the domain and the range of $g^{- 1}$ .

Vehicle Stopping Distance Taking into account reaction time, the distance $d$ (in feet) that a car requires to come to a complete stop while traveling $r$ miles per hour is given by the function

localid="1646198784356" $d (r) = 6.97 r - 90.39$

a. Express the speed $r$ at which the car is traveling as a function of the distance localid="1646198792401" $d$ required to come to a complete stop

b. Verify that localid="1646198797631" $r = r (d)$ is the inverse of localid="1646198802104" $d = d (r)$ by showing that localid="1646198806352" $r (d (r)) = r$ and localid="1646198811055" $d (r (d)) = d$

c. Predict the speed that a car was traveling if the distance required to stop was localid="1646198818232" $300$ feet.

$d (r (d)) = d$

In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer.

$f (x) = \frac{2 x - 3}{x + 4}$

See all solutions

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