We are given approximations of log2, log3, and log5. This allows us to approximate other logarithms. To do so, we will 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
To find the approximation of log6, we should write it in terms of the given logarithms. According to the Product Property, the logarithm of a product is the same as the sum of the logarithms of its factors. Note that
6 is a product of 2 and 3.
log6=log(2* 3)
This means we can rewrite the logarithm as a sum.
log(2* 3)=log2+log3
Now, we can substitute the given values and find log6.
Similarly as in Part A, we will use the Properties of Logarithms to write log15 in terms of the given logarithms. In this case, note that 15 is the product of 3 and 5.
log15 = log(3*5)By the Product Property of Logarithms, we can rewrite it as a sum of the logarithms of factors.
log(3*5) = log3+log5
Let's substitute the given approximations and find log15.
Once again, to find the approximation of log9 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 9 is a square of 3.
log9 = log3^2This means we can rewrite the logarithm as a product.
log3^2= 2log3
Now we can substitute the given approximation for log3 and find log9.
Finally, let's find the approximation of log50. To write it in terms of the given logarithms, we should first find factors of 50. Since 50 is even, we can divide it by 2.
log50=log( 2*25)
By the Product Property, we can write the logarithm as a sum of logarithms.
log(2*25)=log2+log25
Now, note that 25 is a square of 5.
log2+log25=log2+log 5^2
According to the Power Property, we can rewrite the logarithm of a power as a product of the logarithm and the exponent.
log2+log5^2=log2+ 2log5
Finally, we can substitute the given approximations.
