d Apply the Product Property and Quotient Property of Logarithms.
A
a log_3(5m)
B
b log_6(p/m)
C
c See solution.
D
d log(10)
Practice makes perfect
a We want to rewrite the given logarithmic expression using only one logarithm.
log_3(5) + log_3(m)
The given expression contains logarithms with the same base being added together, so let's use the Product Property of Logarithms to rewrite it.
log_b m + log_b n= log_b mn
In this formula, m, n, and b are positive numbers and b≠1. Let's apply these properties to simplify our expression.
log_3(5) + log_3(m)
log_3(m) + log_3(n)=log_3(mn)
log_3(5m)
The equivalent expression using one logarithm is log_3(5m).
b The next expression contains logarithms with the same base and the subtraction.
log_6(p) - log_6(m)
This is why we will use the Quotient Property of Logarithms to rewrite it.
log_b m - log_b n=log_b (m/n)
In this formula, m, n, and b are positive numbers and b≠1. Let's apply these properties to simplify our expression.
log_6(p) - log_6(m)
log_6(m) - log_6(n)=log_6(m/n)
log_3(p/m)
The equivalent expression using one logarithm is log_6( pm).
c We want to rewrite the given expression using only one logarithm.
log_2(r) + 3 log_6(z)
Unfortunately, we cannot write this expression with only one logarithm because the logarithms have different bases. Let's instead adjust our given expression so that we can do it. We can do this in a couple of ways.
First Alternative Exercise
Suppose the bases are the same — for example, let the base for both logarithms be 2.
log_2(r) + 3log_2(z)
The expressions contains logarithms with the same bases, addition, and multiplication in front of one of the logarithms. This is why we are going to use the Power and Product Properties of Logarithms.
Product Property:& log_b mn = log_b m + log_b n
Power Property:& log_b m^r = r log_b m
In these formulas, m, n, and b are positive numbers and b≠1, and r is any real number. Let's apply these properties to simplify our expression.
log_2(r) + 3log_2(z)
m* log_2(a)=log_2(a^m)
log_2(r) + log_2(z^3)
log_2(m) + log_2(n)=log_2(mn)
log_2(rz^3)
The equivalent expression using one logarithm is log_2(rz^3).
Second Alternative Exercise
Now let's solve a similar exercise with different bases that allows us to rewrite it as only one logarithm. Suppose that the base of the second logarithm is 8 instead of 6.
log_2(r) + 3log_8(z)
Since the bases are different, we need change the base of one of the logarithms. Let's use the Change of Base Formula.
log_a(b) = log_c(b)/log_c(a)
This property holds for all positive numbers a,b,c, where b ≠1 and c ≠1. Let's change the base of the second logarithm with c = 2.
log_8( z) = log_2( z)/log_2( 8)
We can substitute this into the expression we want to simplify.
log_2(r) + 3log_8(z) ⇕ log_2(r) + 3* log_2( z)/log_2( 8)
Let's simplify the expression!
The equivalent expression using only one logarithm is log_2(rz).
d This time the given expression contains the logarithms with the same base but uses multiple operations. Recall that when we do not write the base, it is understood to be 10.
log(90)+log(4)-log(36)
We will use both the Product Property and the Quotient Property of Logarithms.
We found that the equivalent expression using only one logarithm is log(10).
Extra
Simplifying the Answer
We can simplify this expression even further and get rid of the logarithm altogether! Recall the definition of a logarithm.
log_b(m) = n ⇔ b^n = m
In our exercise, b = 10 and m = 10. Let's substitute these values into the definition.
log_(10)(10) = n ⇔ 10^n = 10
We know that 10^1=10. Because of this, we can say that log(10) = 1.
