CAIE IGCSE Additional Math 6 logarithmic and exponential functions

Use this Logs and Exponentials hub to move from graph domains and intercepts into logarithm laws and exponential equations solved with logs.

Covered syllabus points

6.1. Use logarithmic and exponential graphs 6.2. Use laws of logarithms 6.3. Solve exponential equations

[Maximum number: 4]

Variables y and x are known to be connected by the relationship $y=A b^{x}$ where A and b are constants. The table shows values of y for certain values of x.

Table for Question 6. Logarithmic and exponential functions question 1 — CAIE IGCSE Additional Math

(a)

Draw the graph of $\lg y$ against x.

Figure for Question (a) — CAIE IGCSE Additional Math

[ 2 ]

(b)

Find an estimate of x when y=1500.

[ 2 ]

[Maximum number: 9]

(a)

Use logarithms to solve the following equation, giving your answer correct to 1 decimal place.
$5^{x-2}=3\times2^{2x+3}$

[ 4 ]

$\log 5^{x-2}=\log3+\log2^{2x+3}$ soi
M1
$(x-2)\log5=\log3+(2x+3)\log2$
M1
$x=\frac{\log3+3\log2+2\log5}{\log5-2\log2}$
A1
x=28.7
A1

(b)

Solve the equation $\log_3(y^2+11)-2=\log_3(y-1)$ .

[ 5 ]

$\log_3\left(\frac{y^2+11}{9}\right)=\log_3(y-1)$
B1
$\frac{y^2+11}{9}=y-1$
M1
$y^2-9y+20=0$
A1
Solves the quadratic
M1
y=4, y=5
A1

[Maximum number: 3]

Solve the equation $5^{2y-1}=6\times 3^y$ , giving your answer correct to 3 decimal places.

$(2y-1)\log 5=\log 6+y\log 3$
oe, or $2y\log 5=\log 30+y\log 3$ oe
OR rearranges $\frac{5^{2y}}{5}=6\times3^y$ and collects powers to $\left(\frac{5^2}{3}\right)^y=30$ oe
M1
$y(2\log5-\log3)=\log6+\log5$
oe, or $y(2\log5-\log3)=\log30$ oe
OR $y=\log_{25/3}30$ oe, or $y\log\left(\frac{5^2}{3}\right)=\log30$ oe
M1 FT if of equivalent difficulty
1.604
A1 cao

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Covered syllabus points

6. Logarithmic and exponential functions question 1

Question (a)

Question (b)

6. Logarithmic and exponential functions question 2

Question (a)

Question (b)

6. Logarithmic and exponential functions question 3