Application of Logs - [Pure 3].

Question 1

If a^x = b, then x = ______.

In terms of log.

Answer 1

x = log_a b.

or x = logb / loga.

Question 2

log_ax + log_ay = ______.

Answer 2

log_ax + log_ay = log_a xy.

Question 3

log_ax - log_ay = ______.

Answer 3

log_ax - log_ay = log_a (x/y).

Question 4

log_axⁿ = ______.

Answer 4

log_axⁿ = nlog_a x.

Question 5

lnx + lny = ______.

Answer 5

lnx + lny = ln xy.

Question 6

lnx - lny = ______.

Answer 6

lnx - lny = ln (x/y).

Question 7

lnxⁿ = ______.

Answer 7

lnxⁿ = nln x.

Question 8

What are the 2 types of Exponential Graphs?

Answer 8

Exponential Growth and Exponential Decay.

Question 9

What is the General Equation for Exponential Growth/ Decay?

Answer 9

y = Ae^bx.
or
P = Ak^t.

Question 10

When using a model such as: P = Ak^t, how do you find ‘A’?

Answer 10

Set t = 0 and you should already have P. k^t will equal 1 and so A will generally equal P when t = 0.

Question 11

What are the Linear Laws?

Answer 11

The relationships between equations such as y = axⁿ or y = kb^x and the linear equation Y = mX + c.

Question 12

Show that y = axⁿ can be written as Y = mX + c.

Answer 12

y = axⁿ
–> log₁₀y = log₁₀axⁿ
–> log₁₀y = log₁₀a + log₁₀xⁿ
–>log₁₀y = log₁₀a + nlog₁₀x
………………………..where………………………..
log₁₀y = Y,
log₁₀a = c,
nlog₁₀x = mX.
–> Y = c + mX.

Question 13

Show that y = kb^x can be written as Y = mX + c.

Answer 13

y = kb^x
–> log₁₀y = log₁₀kb^x
–> log₁₀y = log₁₀k + log₁₀b^x
–>log₁₀y = log₁₀k + xlog₁₀b
………………………..where………………………..
log₁₀y = Y,
log₁₀k = c,
xlog₁₀b = mX.
–> Y = c + mX.

Question 14

On a linear graph, what part of log₁₀y = log₁₀a + nlog₁₀x or log₁₀y = log₁₀k + xlog₁₀b does the y-intercept equal?

Answer 14

log₁₀a or log₁₀k.

Question 15

On a linear graph, what part of log₁₀y = log₁₀a + nlog₁₀x or log₁₀y = log₁₀k + xlog₁₀b does the gradient equal?

Answer 15

nlog₁₀x or xlog₁₀b, where the n or x is the ‘m’ part of the gradient and the log₁₀x or log₁₀b is the ‘x’ part.

Question 16

How do you solve for ‘y’ from a linear graph of lny or log₁₀y against x?

Answer 16

You should get the y-intercept to be lny = a or log₁₀y = a and you should take ‘e’ on both sides to get y = e^a.