Question

How do you evaluate ${{\log }_{7}}\left( -49 \right)$?

Answer 1

Hint: Write the given logarithmic expression in the exponential form i.e., in the form of the powers. Find the power of $7$ for which we get the value $-49$ . We need to find the number which is a real number that satisfies the equation. Also, keep in mind the main definition of a logarithmic function; its conditions, and exceptions to easily solve this problem using pure logic.

Complete step by step solution:
The given expression is ${{\log }_{7}}\left( -49 \right)$
Let us first write this in the form of the powers.
The logarithm of a given constant $y$ is the exponent to which another fixed constant, the base $b$ , must be raised, to produce that constant $y$.
$\Rightarrow {{\log }_{b}}\left( {{b}^{x}} \right)=x$
Which can also be expressed as,
${{\log }_{a}}b=x\Leftrightarrow {{a}^{x}}=b$
Hence substituting the values in our question in this formula we get,
${{\log }_{7}}\left( -49 \right)\Leftrightarrow {{7}^{x}}=-49$
Here we can see that there is no real number in $x$ that satisfies the expression.
No real value of $x$ gives the value which is negative.
Hence it is undefined.
The major definition of a logarithmic function is that,
Logarithms are only defined for $\log x;x>0$ and undefined for $x\le 0$ .
Hence the solution for our expression ${{\log }_{7}}\left( -49 \right)$ is undefined.

Note: Always when a logarithmic expression is given, check if the values inside are greater than zero because logarithmic functions are only defined for the values greater than zero. The same applies to the base also. The base value also must be greater than zero. Students also need to keep in mind that sometimes logarithm is written without a base. In this case, we must assume that the base is 10. It is called the “common logarithm”.