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Exponential Functions

Solve the Equation

A) -3
B) 1
C) 4 and half
D) 3

Exponential Functions Solve the Equation A) -3 B) 1 C) 4 and half D) 3-example-1
User WayneSan
by
2.4k points

2 Answers

13 votes
13 votes

Answer:

  • C) 4 and a half

Explanation:

To solve this question, we can use either of the 2 below given methods:

  1. Rule of exponents
  2. Logarithms

1. Rule of Exponents:


2 ^ { 7-2x } = ( 1 )/( 4 )\\\rightarrow 2 ^ { 7-2x } = ( 1 )/( 2^(2) )

Now, by using the law →
x^(-y) = (1)/(x^(y))...


2 ^ { 7-2x } = ( 1 )/( 2^(2) )\\2 ^ { 7-2x } = 2^(-2)

Now, let's take the exponential values as the base values are equal.


7 - 2x = - 2\\- 2x = - 2 + (-7)\\- 2x = - 9\\\boxed{x = (9)/(2) = 4.5}

2. Logarithms:


2 ^ { 7-2x } = ( 1 )/( 4 )\\\rightarrow2^(-2x+7)=(1)/(4)

Now, take the logarithm of both the sides of the equation.


\log(2^(-2x+7))=\log((1)/(4))

We know that, the logarithm of a number raised to an exponential power is power times the logarithm of the number. So,


\log(2^(-2x+7))=\log((1)/(4)) \\ \rightarrow \left(-2x+7\right)\log(2)=\log((1)/(4))

Now, divide both the sides of the equstion by log (2).


-2x+7=(\log((1)/(4)))/(\log(2))

According to the change of base formula,
(\log(x))/(\log(y)) =
\log_(y)(x). Then,


-2x+7=\log_(2)\left((1)/(4)\right)

By subtracting 7 from both the sides of the equation & then simplifing it further....


-2x=-2-7 \\-2x = - 9\\\boxed{x = (9)/(2) = 4.5}

  • We get the same value by using either of the 2 methods.
  • The value of x = 9/2 or 4.5

_____________

Hope it helps!


\mathfrak{Lucazz}

User Mohit Chandel
by
3.3k points
15 votes
15 votes

Answer:

option c .

Explanation:

Given exponential equation is ,


\longrightarrow 2^(7-2x)= (1)/(4)

As we know that 4 = 2² , so ;


\longrightarrow 2^(7-2x)= (1)/(2^2)

Recall that ,
a^(-m)=(1)/(a^m) .So ;


\longrightarrow 2^(7-2x)= 2^(-2)

Since the bases are equal we can compare the powers as ,


\longrightarrow 7-2x = -2

Subtracting 7 on both sides,


\longrightarrow -2x = -9

Divide both sides by -2,


\longrightarrow x =(-9)/(-2)

Simplify,


\longrightarrow \underline{\underline{ x = 4.5 }}

Hence the correct option is C .

User Binary Nerd
by
3.1k points