Question

3*4^x + 5 * 4^x-3 =197

Answer 1

3*4^x+5*4^x*4^-3=197
4^x(3+5*1/64)=197
4^x *(192+5/64)=197
4^x=197*64/197
4^x=64
4^x=4^3
x=3

Answer 2

To solve the equation 3 * 4^x + 5 * 4^(x-3) = 197, we can start by simplifying the equation.

Let's rewrite 4^x-3 as 4^(x) / 4^3, since dividing by 4^3 is equivalent to subtracting 3 from the exponent.

Now, the equation becomes:

3 * 4^x + 5 * (4^x / 4^3) = 197

To simplify further, we can combine the terms with the same base, 4^x:

(3 + 5/4^3) * 4^x = 197

Now, let's evaluate the term (3 + 5/4^3):

3 + 5/4^3 = 3 + 5/64 = 3 + 5/64 = 192/64 + 5/64 = 197/64

Substituting this back into the equation:

(197/64) * 4^x = 197

To isolate 4^x, we can divide both sides by (197/64):

4^x = 197 / (197/64)

Now, let's simplify the right side of the equation:

4^x = 64

Taking the logarithm of both sides with base 4:

log₄(4^x) = log₄(64)

x * log₄(4) = log₄(64)

Since log₄(4) = 1:

x = log₄(64)

Using a change of base formula, we can rewrite the equation:

x = log₄(64) = log(64) / log(4)

Evaluating this expression:

x = log(64) / log(4) = 2 / 0.6021 ≈ 3.322

Therefore, the solution to the equation 3 * 4^x + 5 * 4^(x-3) = 197 is approximately x = 3.322.