How do you solve #7^(x – 2) = 12x#?

Answer 1

A step in the correct direction!! Perhaps someone else can take it further. I used the Iteration method.

#color(red)("Method 1")#
Plot the straight line graph of #12x#
Plot the curve of #7^(x-2)#
The intersections are where the values for #x# are.

Or try iteration for #x =7^x/588#

#color(red)("Method 2")#
#7^(x-2) -> (7^x)/(7^2)#

so #7^(x-2) = 12x -> 7^x = (7^2)(12)x#

#7^x = 588x#

Taking logs
#x ln(7) = ln(588) + ln(x)#

#x = (ln(588) + ln(x))/(ln(7))#

By iteration

Set seed value as 1
#x ~= 3.988# to 3 decimal places

I could not find the seed for the lower one which is in the region of
#x ~= 0.0016# to 5 dp

Graphs of the log method:

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Answer 2

Anna Allen

To solve (7^{x - 2} = 12x), you would typically use numerical or graphical methods, as there is no algebraic way to solve it directly. One common approach is to use numerical methods such as iteration or approximation techniques. Alternatively, you can graph both sides of the equation and find the intersection point(s) to approximate the solution(s).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.