We are given the values of log_x2, log_x5, and log_x7. To write log_x10 in terms of the given logarithms, let's first recall three useful Properties of Logarithms.
Properties of Logarithms
Name
Product Property
Quotient Property
Power Property
Condition
a>0, b>0, and x ≠1
a>0, b>0, and x ≠1
m>0, b>0, and b ≠1
Property
log_x ab = log_x a + log_x b
log_x ab=log_x a - log_x b
log_b m^p = p log_b m
According to the Product Property, the logarithm of a product is the same as the sum of the logarithms of its factors. Note that
10 is a product of 2 and 5.
log_x10=log_x(2* 5)
This means we can rewrite the logarithm as a sum.
log_x(2* 5)=log_x2+log_x5
Now, we can substitute the given values and find log_x10.
Same as in Part A, we will use the Properties of Logarithms. According to the Power Property, the logarithm of a power is the product of the logarithm and the exponent. Note that 49 is a square of 7.
log_x49 = log_x7^2This means we can rewrite the logarithm as product.
log_x7^2= 2log_x7
Now we can substitute the given value for log_x7 and find log_x49.
Once again, to find log_x50 we will use the Properties of Logarithms. Before we do that, we should find factors of 50. Since 50 is even, it is divisible by 2.
log_x50=log_x( 2*25)
By the Product Property, the logarithm of a product is the same as a sum of the logarithms of its factors.
log_x(2*25)=log_x2+log_x25
Now, note that 25 is a square of 5.
log_x2+log_x25=log_x2+log_x 5^2
According to the Power Property, we can rewrite the logarithm of a power as a product of the logarithm and the exponent.
log_x2+log_x5^2=log_x2+ 2log_x5
Finally, we can substitute the given values.
Finally, let's find log_x56. To do so we should first find factors of 56. Note that 56 is divisible by 7.
log_x57=log_x( 7*8)
By the Product Property, we can write the logarithm as a sum of logarithms.
log_x(7*8)=log_x7+log_x8
Now, note that 8 is a power of 2.
log_x7+log_x8=log_x7+log_x 2^3
This allows us to apply the Power Property.
log_x7+log_x2^3=log_x7+ 3log_x2
Let's substitute the given values.
