6.1

Given the equation \(7^x = 2^{x+1}\).

Use logarithms to solve the equation, giving the value of \(x\) correct to \(3\) significant figures.

\(7^x = 2^{x+1}\\ \frac{7^x}{2^x} = 2\\ (3.5)^x = 2\\ x=\log_{3.5}2\approx0.544\)

6.2

Given the equation \(3^{2x+1} = 5^{200}\).

Use logarithms to solve the equation, giving the value of \(x\) correct to \(3\) significant figures.

\(3^{2x+1}=5^{200}\\ 2x+1=\log_3(5^{200})\\ 2x=200\log_35-1\\ x=100\log_35-0.5\approx146.497-0.5=145.997 \)

6.3

Given the equation \(2^{4x-1} = 3^{5-2x}\).

Solve the equation, giving your answer in the form \(\log_ba\).

\(2^{4x-1} = 3^{5-2x}\\ \frac{4^{2x}}{2}=\frac{3^5}{3^{2x}}\\ 12^{2x}=2\times243\\ 144^x=486\\ x=\log_{144}486\)

6.4

Given equations \(\log_{10}x+\log_{10}y=\log_{10}3\) and \(\log_{10}(3x+y)=1\).

Sotve them simultaneously. Choose possible values of \(x\) and \(y\).

 \(\left\{ \begin{aligned} &\log_{10}x+\log_{10}y=\log_{10}3\\ &\log_{10}(3x+y)=1 \end{aligned} \right.\\ \left\{ \begin{aligned} &x>0,y>0\\ &\log_{10}(xy)=\log_{10}3\\ &3x+y=10\\ \end{aligned} \right.\\ \left\{ \begin{aligned} &x>0,y>0\\ &xy=3\\ &y=10-3x \end{aligned} \right.\)

\(x(10-3x)=3\\ 10x-3x^2=3\\ 3x^2-10x+3=0\\ D=10^2-4\times3\times3=100-36=64\\ x=\frac{10\pm8}{6}\)

So, \(x=3\) or \(x=\frac13\). Therefore, \(y=1\) or \(y=9\) respectively.

6.5

Given the equation \(7^{w-3}-4=180\).

Use logarithms to solve the equation, giving the value of \(x\) correct to \(2\) significant figures.

\(7^{w−3}−4=180\\ 7^{w−3}=184\\ w-3=\log_7184\\ w=\log_7184+3\approx2.68+3=5.68\)

6.6

The curve \(y=2^x-3\) passes through the point \((p,62)\).

Use logarithms to find the value of \(p\), correct to 3 significant figures.

To find \(p\) we should solve the equation

\(62=2^p-3\\ 2^p=65\\ p=\log_265\approx0.301\)

6,7

Given the equation \(\log_2x+2\log_23=\log_2(x+5)\).

Find \(x\).

\(\log_2x+2\log_23=\log_2(x+5)\\ \log_2x+\log_29=\log_2(x+5)\\ \log_29x=\log_2(x+5)\\ \left\{ \begin{aligned} &x>0\\ &9x=x+5 \end{aligned} \right.\\ \left\{ \begin{aligned} &x>0\\ &8x=5 \end{aligned} \right.\\ x=\frac58=0.625\)

6.8

The point \(P\) on the curve \(y=9^x\) has \(y\text{-coordinate}\) equal to \(150\).

Use logarithms to find the \(x\text{-coordinate}\) of \(P\), correct to \(3\) significant figures. 

To find \(x\) e should solve the equation

\(150=9^x\\ x=\log_9150\approx2.28\)

6.9

Given that \(\log_x(5y+1)-\log_x3=4\).

Express \(y\) in terms of \(x\).

\(\log_x(5y+1)-\log_x3=4\\ \log_x\frac{5y+1}{3}=4\\ \frac{5y+1}{3}=x^4\\ 5y+1=3x^4\\ 5y=3x^4-1\\ y=\frac35x^4-\frac15\)

6.10

Given the equation \(5^{3w-1}=4^{250}\).

Use logarithms to solve the equation, giving the value of \(w\) correct to \(3\) significant figures.

\(5^{5w-1}=4^{250}\\ 5w-1=\log_5(4^{250})\\ 5w=250\log_54+1\\ w=\frac{250}{5}\log_54+\frac15=50\log_54+0.2\approx43.068+0.2=43.268\)