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I have posted this question previously however I am asking now specifically for question (d) (im stuck)

I know this is very basic stuff but I don't know how to find the answer to question (d) in this. The previous parts to this question are below (in case they are relevant).The questions are about the log function (in base 10):

$f(x)=3log10(10x+4)+2$

a) The largest domain of $f$ is $x>a$ for some $a$, therefore what is the value of $a$?

b) What is $f(2.3)$?

c) What is $f(x)=8.2803$ for $x$?

d) Say $b$ is a number for which both $b$ and $1000b+1998/5$ are in the domain of $f(x)$. Find the simplest possible expression for $f(1000b+1998/5)−f(b)$.

For question (a) I calculated "a" to be $−0.4$.

For question (b) I calculated it to be $6.29409$ by formulating it as $f(x)=3log10(10(2.3)+4)+2$ and following that through.

For question (c) I calculated it to be $12.0$ by formulating it as $8.2803=3log10(10(x)+4)+2$ and solving for x.

I, however, don't know how to get question (d). Can someone show me the process of how to do it?

user1030161
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1 Answers1

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So $f(x) = 3\log_{10}(10x + 4) + 2$, hence \begin{align*} f\left(1000b + \frac{1998}5\right) &= 3\log_{10}(10^4b + 3996 + 4) + 2\\ &= 3\log_{10}\bigl(10^3(10b + 4)\bigr) + 2\\ &= 3\bigl(3 + \log_{10}(10b + 4)\bigr) + 2\\ &= 3\log_{10}(10b+ 4) + 9 + 2\\ &= f(b) + 9\\ \iff f\left(1000b + \frac{1998}5\right) - f(b) &= 9 \end{align*}

martini
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