Question

Find the value of x, when: ${{(-2)}^{x}}=-128$.

Answer 1

Hint: To solve this problem, we should be aware about the basic properties of logarithm. Basically, if we have an equation,
${{a}^{x}}=b$
Then, we have, x = ${{\log }_{a}}b$. Here, a and b are any two numbers. Thus, we will use this property to solve the above mentioned problem.

Complete step-by-step answer:

Before solving, we will first understand the basic definition of logarithm. Logarithm is another way of thinking about exponents. For example we take an example of 3 raised to the ${{3}^{rd}}$ power, this would be equal to $3\times 3\times 3$=27. Now, in this case we were aware about the power to which 3 was to be raised. However, suppose we are asked what power should 3 be raised for the answer to be equal to 27, then we make use of logarithms. Equation in hand would be-
${{3}^{x}}=27$
Thus, to find x, we use logarithms here, applying logarithm on LHS and RHS, we get,
${{\log }_{3}}({{3}^{x}})={{\log }_{3}}(27)$
Now, in general, we should know that, ${{\log }_{a}}{{a}^{x}}$=x (for any positive number a). Applying this property, we get,
x=${{\log }_{3}}27$ (this has a value of 3)
Now, coming onto the actual problem, we have,
${{(-2)}^{x}}=-128$
Since, for logarithm, we need to have a positive number, inside the logarithm, we do some manipulations, we get,
${{(-1)}^{x}}{{(2)}^{x}}=(-1)\times 128$
Now, we need to have a positive term when we take logarithm, thus, we need to have,
$-{{1}^{x}}$ = -1
For this to happen, x should be an odd number. Thus, we assume x to be an odd number, we further solve (if x turns out to be even number, we will have no solutions for this equation),
${{(2)}^{x}}=128$ (Since, now $-{{1}^{x}}$ = -1 due to assumption of x as odd number)
Thus, x = ${{\log }_{2}}128$
x=${{\log}_{2}}2^7$
x = 7 (which is an odd number)
Thus, our assumption is correct and the equation does have a solution.
Hence, the correct answer to the equation is x = 7.

Note: In the expression, x = ${{\log }_{a}}b$, it is important to note that b should be a positive number. Further, ‘a’ should also be a positive number excluding 1. Thus, while solving a question these conditions must be kept in mind to avoid any errors. Thus, in this question, we had to meet this condition by taking x as an odd number, else the value of b would have been negative (that is, -128).