###### 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.

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