How To Evaluate Logarithms With Square Root Bases

The logarithm of a number identifies the power that a specific number, referred to as a base, must be raised to produce that number. It is expressed in the general form as log a (b) = x, where a is the base, x is the power that the base is being raised to, and b is the value in which the logarithm is being calculated. Based on these definitions, the logarithm can also be written in exponential form of the type a^x=b. Using this property, the logarithm of any number with a real number as the base, such as a square root, can be found following a few simple steps.

Step 1

Convert the given logarithm to exponential form. For example, the log sqrt(2) (12) = x would be expressed in exponential form as sqrt(2)^x = 12.

Step 2

Take the natural logarithm, or logarithm with base 10, of both sides of the newly formed exponential equation.
log( sqrt(2)^x) = log (12)

Step 3

Using one of the properties of logarithms, move the exponent variable to the front of the equation. Any exponential logarithm of the type log a ( b^x) with a particular "base a" can be rewritten as xlog a (b). This property will remove the unknown variable from the exponent positions, thereby making the problem much easier to solve. In the previous example, the equation would now be written as: xlog(sqrt(2)) = log (12)

Step 4

Solve for the unknown variable. Divide each side by the log(sqrt(2)) to solve for x: x=log(12)/log(sqrt(2))

Step 5

Plug this expression into a scientific calculator to get the final answer. Using a calculator to solve the example problem gives the final result as x = 7.2 .

Step 6

Check the answer by raising the base value to the newly calculated exponential value. The sqrt(2) raised to a power of 7.2 results in the original value of 11.9, or 12. Therefore, the calculation was done correctly:

sqrt(2)^7.2 =11.9