Answer :
To express [tex]\log_{10} 30[/tex] and [tex]\log_{10} 450[/tex] in terms of [tex]\log_{10} 2[/tex], [tex]\log_{10} 3[/tex], and [tex]\log_{10} 5[/tex] to any base, we can use the properties of logarithms. Specifically, we'll use the product rule for logarithms which states that [tex]\log_b(xy) = \log_b x + \log_b y[/tex].
Let's start with [tex]\log_{10} 30[/tex]:
- First, we'll factor 30 into its prime factors:
[tex]30 = 2 \times 3 \times 5[/tex] - Using the product rule, we can rewrite [tex]\log_{10} 30[/tex] as:
[tex]\log_{10} 30 = \log_{10} (2 \times 3 \times 5) = \log_{10} 2 + \log_{10} 3 + \log_{10} 5[/tex]
Now, let's move on to [tex]\log_{10} 450[/tex]:
- Factor 450 into its prime factors:
[tex]450 = 2 \times 3^2 \times 5^2[/tex] - Using the product rule again, rewrite [tex]\log_{10} 450[/tex]:
[tex]\log_{10} 450 = \log_{10} (2 \times 3^2 \times 5^2) = \log_{10} 2 + 2 \log_{10} 3 + 2 \log_{10} 5[/tex]
By expressing these logarithms in terms of [tex]\log_{10} 2[/tex], [tex]\log_{10} 3[/tex], and [tex]\log_{10} 5[/tex], we've used the properties and identities of logarithms to decompose them into simpler parts. This method of decomposition makes it possible to convert complex logarithmic expressions into sums of simpler logarithms.