# Simultaneous equations with logarithms?

Given that $$a$$ and $$b$$ are positive constants and $$a>b$$, solve the simultaneous equations.

$$a+b=13$$
$$\log_6 a +\log_6 b= 2$$

I have tried doing this but I can’t figure out what to do first. I am thinking that you take logs of equation $$1$$ but I’m not sure.

• Hint: a sum of logs is... – Sean Roberson Sep 22 '18 at 8:18
• $\log a+\log b=\log {ab}$ – For the love of maths Sep 22 '18 at 8:20
• By learning a little MathJax/LaTeX (there are some links in the help center) you can format math to look much better, and avoid having to write comments about what is subscripts). – Henrik supports the community Sep 22 '18 at 8:26

$$\log_6a+\log_6b=\log_6(ab)$$

$$\log_6x=2$$ if and only if $$x=6^2=36$$

Can you go on now, without peeking at the spoiler?

$$\begin{cases}a+b=13\\ab=36\end{cases}$$ has an obvious solution

It comes down to finding two numbers, given their sum $$s$$ and their product $$p$$ : $$\begin{cases}a+b=13\\ \log_6 a+\log_6 b=2\end{cases}\iff\begin{cases}a+b=13\\ \log_6ab=2\end{cases}\iff\begin{cases}a+b=13\\ ab=6^2=36\end{cases}$$ so $$a$$ and $$b$$, by the theory of quadratic equations, are the roots of $$x^2-sx+p=x^2-13x+36=0.$$ You can find the roots applying the rational roots theorem.

• Thanks. How do you know that log a +log b = 36? – Eleanorm Sep 23 '18 at 7:41
• Just because $\;\log_6(ab)=2\iff ab=6^2$. – Bernard Sep 23 '18 at 10:23

Taking the logarithm of a sum leads you nowhere. But taking the antilogarithm yields a product.

$$6^{\log_6a+\log_6b}=6^{\log_6a}6^{\log_6b}=ab=6^2=36.$$

You have reduced to a problem where the sum and product of two numbers is known. By the Vieta formulas, these are the solutions of the quadratic equation

$$x^2-13x+36=0.$$

Hint: Writing $$\frac{\ln(a)}{\ln(6)}+\frac{\ln(b)}{\ln(6)}=2$$ with the first equation $$b=13-a$$ we get $$\ln(a)+\ln(13-a)=2\ln(6)$$ Using the law of logarithm we get $$\ln(a(13-a))=2\ln(6)$$ and $$a(13-a)=e^{2\ln(6)}$$ Can you finish?