Definition of Logarithm: The Meaning of log base A of x
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Determine log base 16 of one eigth.
Equivalent Logarithmic and Exponential Equations
Let A be any positive real number other than 1 and let x be any positive real number.
Then the logarithm, base A, of x is the power to which the base A must be raised in order to get x
The logarithm, base A, of x is the abbreviated log base A of x
For example, the logarithm, base 3, of 81 is abbreviated log base 3 of 81.
From the definition of logarithm, we know that log base 3 of 81 is the exponent to which we must raise the base, 3, in order to get 81. Sometimes, as in this case, the numerical value of a logarithm is easy enough to calculate mentally. Consider the first few positive integer powers of 3.
3 to the first equals 3
3 squared equals 9
3 cubed equals 27
3 to the fourth equals 81
We can see that the exponent we need is 4.
Therefore, log base 3 of 81 equals 4 because 3 to the fourth is 81
What is log base 4 of 16?
What is log base 9 of 3?
To what power must we raise the base, 16, in order to get the number one eighth?
Let x be the exponent such that 16 raised to this exponent is one eighth;.
16 to the x = one eighth;
2 to the fourth to the power of x = 2 to the power of negative 3
Substitute 2 to the fourth for 16 and 2 to the negative third for one eighth.
2 to the power of 4 x = 2 to the power of negative 3
Use the Power Rule for Exponents.
4 x = negative 3
Use the Principle of Exponential Equality.
x = negative three fourths;
Divide both sides by 4.
We have determined that the power of 16 that equals one eighth is negative three fourths.
In other words log base 16 of one eighth = negative three fourths
Check:
16 to the power of negative three fourths = 1 over 16 to the power of negative three fourths = 1 over the quantity 4 root 16 to the power of 3 = 1 over 2 to the power of 3 = 1 over 8
What is log base 9 of 27?
As you can see, each value of a logarithm is determined by knowing facts about powers.
log base 3 of 81 equals 4 because 3 to the fourth equals 81
log base 4 of 16 equals 2 because 4 squared equals 16
log base 9 of 3 equals one half because 9 to the power of one half equals 3
log base 16 of one eighth equals negative three fourths because 16 to the negative three fourths equals one eighth
Pairs of equations such as these, that are always either both true or both false, are called equivalent equations.
The logarithmic equation log base A of x = y is equivalent to the exponential equation A to the power of Y = x.
If either of these equations is true, so is the other one.
For example, the exponential equation that is equivalent to log base 5 of 25 = 2 is 5 squared = 25
What is the exponential equivalent to log base 2 of 64 = 6?
What is the exponential equivalent to log base 8 of one fourth = negative two thirds?
What is the logarithmic equivalent to 5 cubed = 125?
What is the logarithmic equivalent to A to the fourth = v?
Being able to convert a given exponential equation to the equivalent logarithmic equation and vice versa is often very important in solving equations involving logarithms or exponents.
Practice writing equivalent equations of these types until you are completely sure about what goes where.
Remembering the basic definition of logarithm will help make everything clear.
The base A logarithm of the number x is the power of A that produces x
So if log base A of x = y, it's because A to the Y = x.
These equations are equivalent!