log(8)+log(125)
According to the Product Property of Logarithms, the sum of logarithms with the same base can be rewritten as a logarithm of a product.
log m+log n=log( m* n)Let's apply the above property to our case.
Now, to evaluate log(1000) let's write a logarithmic equation.
log(1000)=x
In order to solve this equation, we can rewrite the logarithm as an exponential equation using the definition of a logarithm.
log_b m=n ⇔ m= b^n
This relationship tells us that the logarithm n is the exponent to which b must be raised to get m. The base of a common logarithm is 10. Therefore, we have that log1000=log_(10)1000. In our case, x is the exponent to which 10 must be raised to get 1000.
log_(10) 1000=x ⇔ 1000= 10^x
Finally, we will solve the exponential equation.
This time we are given a single logarithm with the base 25. To evaluate it, let's first write a logarithmic equation.
x=log_(25)(125)
Next, using the definition of a logarithm we can convert it to an exponential equation.
Definition:& n=log_b m &&⇔ b^n= m
Equation:& x=log_(25)( 125) &&⇔ 25^x= 125
Note that in the obtained exponential equation both sides are powers of 5. Therefore, we can rewrite the expressions to have a common base.
Now we have two equivalent expressions with the same base. Recall that, if both sides of the equation are equal, the exponents must also be equal.
5^(2x)=5^3 ⇔ 2x= 3
This allows us to find x.
As in Part A, we are given a sum of common logarithms. However, note that the first term is multiplied by 12.
1/2log(25)+log(20)
According to the Power Property of Logarithms, a logarithm multiplied by a constant is the same as the logarithm of a number raised to that constant.
1/2log(25)+log(20)=log(25^(12))+log(20)Raising a number to the power of 12 is the same as taking the square root of the number. Let's simplify the expression.
Now we need to evaluate a single logarithm. To do so, let's write a logarithmic equation.
x=log(100)
Using the definition of a logarithm, we can convert it to an exponential equation.
Definition:& n=log_b m &&⇔ b^n= m
Equation:& x=log( 100) &&⇔ 10^x= 100
Finally, let's solve the exponential equation.
7^(log_7(12))We are given a power of 7, where the exponent is a logarithm with the base 7. According to the Inverse Properties of Logarithms, a power and a logarithm with the same base undo each other.
b^(log_b( a)) = a
This means that the given expression simplifies to 12.
7^(log_7( 12))= 12
