A detailed explanation of the exponential function and logarithmic function. These documents will ensure a thorough and detailed explanation regarding this topic. The documents were created to make the topic easy to understand.
Recall: An exponential function 𝑦 = 𝑎. 𝑏𝑥 is defined for 𝑏 > 0; 𝑏 ≠ 1.
(A revision of Exponential functions can be found on pages 55-58 of your text book)
Considering only exponential functions with 𝑎 = 1 for the moment, the inverse function of
𝑦 = b𝑥 is 𝑥 = by, but we do not yet have the mathematics to write 𝑥 = by in the form
𝑦 =…
The logarithmic function is a new function used to describe the inverse of the exponential
function i.e. we can use logarithms (called logs for short) to make the exponent the subject
of the equation.
𝑦 = log𝑏𝑥 means exactly the same as 𝑥 = 𝑏𝑦
e.g.1 Consider the function 𝑦 = 2𝑥:
function: 𝑦 = 2𝑥:
Inverse function (with 𝑥 the subject): 𝑥 = 2𝑦
Logarithmic form (with 𝑦 the subject): 𝑦 = log2 𝑥
In function notation:
Function 𝑓(𝑥) = 2𝑥
Inverse function 𝑓−1(𝑥) = log2𝑥
As graphs:
y = log2 x
Note:
The graphs reflect in
the line 𝑦 = 𝑥.
y = 2x The 𝑦-intercept of
𝑦 = 2𝑥 becomes the
𝑥-intercept of
𝑦 = log2𝑥.
The asymptote of
𝑦 = 2𝑥 is the 𝑥-axis.
The asymptote of
𝑦 = log2𝑥 is the
𝑦-axis
y=x
1
, e.g.2 Function: f ( x )=¿𝑦 i.e. y=¿
Inverse function (with 𝑥 the subject): 𝑥 = ( 1)
2
Logarithmic form (with 𝑦 the subject): 𝑦 = 𝑙𝑜𝑔1 𝑥
2
In function notation: 𝑓−1(𝑥) = 𝑙𝑜𝑔1 𝑥
2
As graphs:
y=¿
y=x Note:
The graphs reflect in
the line 𝑦 = 𝑥.
The 𝑦-intercept in
1 𝑥
𝑦 = ( ) becomes the
2
𝑥-intercept of
𝑦 = log1𝑥.
2
The asymptote
1 𝑥
of
𝑦
2
= ( ) is the 𝑥-axis.
The asymptote
of
𝑦 = log1𝑥 Is the
𝑦 = log1 𝑥 2
2 𝑦-axis
Let’s take a closer look at LOGARITHMS
We need to be able to convert from exponential form to log form and vice versa.
LOG FORM log𝑎𝑏 = 𝑥
number bas Logarithm/exponent
e
EXPONENTIAL FORM 𝑎𝑥 = 𝑏
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller kalebroodt. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.69. You're not tied to anything after your purchase.
