Exponentials and logarithms
Introduced| English | Meaning |
|---|---|
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | A function in which the variable appears in an exponent, such as a fixed positive base raised to that variable. |
| exponent/ekˈspəʊnənt/ | A value indicating the power to which a base is raised. |
| natural logarithms/ˈnætʃərəl ˈlɒɡərɪθəmz/ | The logarithm to base e, the inverse of the exponential function with base e. |
| compound interest/ˈkɒmpaʊnd ˈɪntrest/ | Interest calculated on the original principal together with previously accumulated interest. |
| exponential decay/ˌekspəˈnenʃl dɪˈkeɪ/ | A decrease in which a quantity is multiplied by the same factor, between zero and one, over equal intervals. |
Repeated multiplication creates a model
- An exponential function 指数函数 such as $y=a^x$ has the variable in the exponent 指数, with $a>0$ and $a\ne1$.
- Its values grow for $a>1$ and decay for $0<a<1$. Equal time steps multiply by the same factor; they do not necessarily add the same amount.
Exponential growth against a straight line
$y = a^x$
A constant factor always overtakes a constant amount, however steep the line.
Use a logarithm as an inverse
- $\log_a y=x$ means $a^x=y$. Real logarithms need a positive argument and a positive base different from 1.
- Natural logarithms 自然对数 use base e. The graph $y=\ln x$ passes through $(1,0)$ and has vertical asymptote $x=0$; exponential graphs have y-intercept 1 and horizontal asymptote $y=0$.
What is log₁₀ 1000?
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It asks what power of 10 gives 1000, and 10³ = 1000.
Apply laws only within their domain
- For positive m and n with the same valid base, $\log(mn)=\log m+\log n$ and $\log(m/n)=\log m-\log n$.
- The power law $\log(m^k)=k\log m$ brings an exponent into a product. There is no corresponding law splitting the logarithm of a sum.
Which law lets you solve for an unknown that sits in an exponent?
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The power law brings k down to the front, where it can be divided out. The fourth option is not a law at all.
log(m + n) equals log m + log n.
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The addition law applies to a product inside the logarithm, never to a sum.
State the time and financial assumptions
- Compound interest 复利 uses $A=P(1+r)^n$ for a stated rate per period and no other cash flows. The rate r is a fraction, not the percentage numeral.
- Exponential decay 指数衰减 uses a factor between 0 and 1. A continuous-time threshold can differ from the first qualifying whole-period sample; real processes need evidence before adopting a model.
5000 grows at 4% a year. After how many WHOLE years does it first exceed 7000?
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n > log 1.4 ÷ log 1.04 = 8.58, and years are whole, so the answer is 9.
Which base gives exponential decay?
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A base between 0 and 1 shrinks the value at every step. A base of 1 never changes it.
In one English sentence, say when you would use a logarithm to solve an equation.
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Example: "I take logarithms when the unknown is in the exponent, because the power law brings it down."
A whole-year threshold. With $A(n)=500(1.04)^n$, the condition $A(n)>600$ gives $n>\ln(600/500)/\ln1.04\approx4.6486$. The first whole year is 5. Check years 4 and 5 to verify the strict inequality.
Check equation and sketch domains separately. In $\ln(x-1)+\ln2=\ln10$, the equation needs $x>1$ and has solution 6. The separate basic graph $y=\ln x$ is defined for $x>0$. Sheet 1.6 practises both equations and graph features.
A positive exponential decay model stays above zero at every finite time. A physical quantity measured as zero may reflect detection limits or failure of the model, rather than an exact zero predicted by the formula.