Formulas for Real Exponents with Logarithms
In this lesson, extension of formulas for positive
number c raised to fractional exponents implies a formula for real
exponents. Formulas for more exponentials functions result. The latter
are inverse functions to logarithms to base c with c = 10 providing the
common logarithm. This lesson describes all and how. Why the
exponential function exp(x) equals ex is explained
here.
From the previous lesson, recall
\[b^{\frac pq } =\exp\left(\frac pq \cdot\ln(b )\right) \] whenever b is
positive and $\frac pq$ is any rational number. Now the change of
notation $x =\frac pq$ gives \[b^x =\exp (x\cdot\ln(b ) ) \] whenever $x$
is a rational number. The function \[ f(x) =\exp (x\cdot\ln(b ) ) \] is a
continous extension of the definition of powers to rational exponents to
powers with real exponent. So we put the power \[ b^x =\exp (x\cdot\ln(b
) ) \] and read it as based b raised to the exponent x. The latter
provides a computational definition of the exponential function $b^x$
with base b > 0. The letter a may be used in place of b below.
Clearly,
\[ \ln(b^x) = \ln( \exp (x\cdot\ln(b ) ) ) = x\cdot \ln(b) \]
Earlier, we introduced the natural number e =e = exp(1) = 2.718281828...
[ irrational] as the solution of $\ln(e) = 1.$ Thus for the natural
number e, , the natural logarithm of e has ln (e) = 1 The latter property
now explains why \[e^x =\exp(x \ln(e)) = \exp(x) \] Due to the equality,
present-day calculators offer an $e^x$ button for the calculation of the
natural exponential function $\exp(x)$ = the inverse function for the
natural logarithm. Therefore
Properties of Exponentials
These properties are immediate consequences of the definition - applied
once or twice - and the properties of the natural logarithm $\ln(x)$, and
its inverse the - natural- exponential function $\exp(x)$.
Exponential of an Exponential
Now for b > 0 \begin{eqnarray*} (b^x)^y &=& \exp( y \ln(b^x))
\\ &=& \exp( y (x \ln(b )) \\ & =& \exp (yx\ln(x)) \\
& = & b^{yx} \\ & = & b^{xy} \end{eqnarray*} Thus
\[(b^x)^y =(b^{xy} = b^{yx} = (b^y)^x \]
Product of Exponentials - Exponential Sum Rule
\begin{eqnarray*} a^x \cdot a^y &=& \exp(x ln(a)) · \exp(y ln(a)
\\ &=& \exp(x \ln(a)+y \ln(a)) \\ &=& \exp( (x +y )\ln(a)
) \\ &=& a^{x+y} \end{eqnarray*}
Therefore $a^x$ has the exponential property
\[ a^x \cdot a^y = a^{x+y}\] Here $x$ may be whole number, a fraction
$\frac pq$ where p and q are nonzero integers, or more generally, any
real number. The foregoing logarithmetic and exponential function imply
most, if not all, the properties of exponential functions $a^x$.
Connections with logarithms to base a = c > >0
For positive numbers $c \ne 1$, we have or may put \[ \log_c(x) =
\frac{\ln(x)}{\ln(c)}\] with the condition $c \ne 1$ to avoid division by
zero as $\ln(1) =0$
Now \begin{eqnarray*} log_a(a^x) & =& \frac{\ln(a^x)}{\ln(a)} \\
& =& \frac{\ln(\exp(x ln(a)))}{\ln(a)} \\ & =& \frac{ x
ln(a) }{\ln(a)} \\ &= & x \end{eqnarray*} for all real numbers x.
Further, \begin{eqnarray*} a^{log_a (x)} & =& \exp (log_a (x)
\cdot \ln(a) ) \\ & =& \exp \left(\frac{\ln(x)}{\ln(a)} \cdot
\ln(a) \right) & =& \exp(\ln(x)) \\ &= & x
\end{eqnarray*} Thus for a positive and not equal to 1, the two functions
\[g(x) = \log_a(x)\] defined for all positive real numbers x, and \[h(x)
= a^x \] defined for all real numbers x are an inverse function pair,
with the domain of each being the range of the other.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
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protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
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Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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