Use the definition of logarithm to simplify each expression.
log10 10,000
log10 10
log10 10−4
x=log1010000
10x=10000
therefore x=4, or
log1010000=4
To simplify the given expressions using the definition of logarithm, we need to rewrite them in exponent form.
1. log10 10,000:
The logarithm of a number to the base 10 is the exponent to which 10 must be raised to obtain that number. In this case, 10 raised to what exponent equals 10,000?
Rewriting 10,000 in exponent form, we have 10^4. So, log10 10,000 simplifies to 4.
2. log10 10:
Using the same logic as above, 10 raised to what exponent equals 10? Since 10^1 equals 10, the logarithm is equal to 1.
3. log10 10−4:
To simplify this expression, we need to rewrite it in exponent form. However, we have a negative value in the logarithm.
To deal with negative logarithms, we can use the property that log10 b = -log10 (1/b). Applying this property, we have:
log10 10−4 = -log10 (1/10^4)
= -log10 (1/10,000)
Now, we can simplify it using the knowledge gained from the first expression:
= - (-4)
= 4
So, log10 10−4 simplifies to 4.
Therefore, the simplified expressions using the definition of logarithm are:
1. log10 10,000 = 4
2. log10 10 = 1
3. log10 10−4 = 4