Light Magnitude Limit of a Telescope

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Questions and Answers

Explain how powers (numbers with indices) can be used to represent repeated multiplication of the same factor. Provide an example.

Powers provide a shorthand way of writing repeated multiplication. For example, $2 \times 2 \times 2$ can be written as $2^3$, and $5 \times x \times x \times x \times x$ can be written as $5x^4$.

State the index law for multiplication and explain when it is applicable.

The index law for multiplication states that $a^m \times a^n = a^{m+n}$. It is only applicable when the powers have the same base.

Explain the index law for division, its formula, and when it can be used.

The index law for division states that $a^m \div a^n = a^{m-n}$. It is applicable only when dividing powers with the same base.

Describe the index law for the power of a power. Give an example.

The index law for the power of a power states that $(a^m)^n = a^{m \times n}$. For example, $(x^2)^3 = x^{2 \times 3} = x^6$. Signup and view all the answers

Explain the index law for brackets and provide an example.

The index law for brackets states that $(a \times b)^m = a^m \times b^m$. For example, $(2a)^3 = 2^3 \times a^3 = 8a^3$. Signup and view all the answers

State the index law for fractions and provide an example.

The index law for fractions states that $(\frac{a}{b})^m = \frac{a^m}{b^m}$. For example, $(\frac{x}{y})^2 = \frac{x^2}{y^2}$. Signup and view all the answers

Explain the zero index law and provide an example.

The zero index law states that any non-zero number raised to the power of zero equals 1, i.e., $a^0 = 1$ (where $a \neq 0$). For example, $5^0 = 1$. Signup and view all the answers

Express the relationship between a negative index and division.

A negative index indicates division. Specifically, $a^{-n} = \frac{1}{a^n}$. Signup and view all the answers

Why is it important to be able to rewrite expressions involving negative indices with positive indices?

Rewriting expressions with positive indices simplifies calculations and makes it easier to understand the value represented by the expression. Signup and view all the answers

Explain practical scenarios where calculations involving negative powers of 2 are used.

Calculations involving negative powers of 2 are used to determine the quantity remaining after multiple halvings. Applications include radioactive waste management and diagnostic medicine. Signup and view all the answers

How do we rewrite an expression containing roots to use positive indices?

To rewrite any expression containing roots to use positive indices, simply rewrite the root as the equivalent fractional power. Signup and view all the answers

Explain how to rewrite a number using scientific notation. What is the general form of a number in scientific notation?

To write a number in scientific notation, express it in the form $a \times 10^m$, where $1 \le a < 10$ and $m$ is an integer. Signup and view all the answers

When are significant figures counted from in scientific notation? What role to they play in representing numbers in scientific notation?

Significant figures are counted from left to right, starting at the first non-zero digit. They indicate the precision with which a number is known. Signup and view all the answers

If two numbers are multiplied, how many significant figures should be quoted in the final result?

The result should be rounded to the same number of significant figures as the number with the fewest significant figures. Signup and view all the answers

Give some examples of careers that use scientific notation?

Users of scientific notation include astronomers, space scientists, chemists, engineers, environmental scientists, physicists, biologists, lab technicians, and medical researchers. Signup and view all the answers

What determines whether an equation can be solved by expressing both sides using a common base?

The equation can be solved using a common base if both sides can be expressed as powers of the same base. The index laws can then be used to determine the unknown value. Signup and view all the answers

State the general form of a simple exponential equation?

A simple exponential equation is of the form $a^x = b$, where $a > 0$, $b > 0$ and $a \neq 1$. Signup and view all the answers

What feature must be present for an exponential to be solved by equating powers?

The equation must have all base terms as a common base for its powers, so that the indices can be directly equated. Signup and view all the answers

Explain the meaning of an asymptote for an exponential function.

An asymptote is a line that a curve approaches but never touches. The exponential curve approaches the asymptote but never intersects it, meaning it gets infinitely close but never reaches it. Signup and view all the answers

What are the coordinates and the equation of the asymptote?

The y-intercept has coordinates (0, 1), and y = 0 is the equation of the asymptote. Signup and view all the answers

Describe the shape, key features of an exponential curve.

The simplest exponential curves have a y-intercept at (0, 1) where graphs of the form y=ax, a >0 have an asymptote with equation y=0 (the x-axis). Signup and view all the answers

State the basic exponential formula? When is exponential growth possible?

The basic exponential formula is $A = ka^t$, and with $A$ as amount, $k$ as the initial amount and $t$ as the time. Exponential growth occurs when $a > 1$. Signup and view all the answers

State the basic exponential formula? When is exponential decay possible?

The basic exponential formula is $A = ka^t$, and with $A$ as amount, $k$ as the initial amount and $t$ as the time. Exponential decay occurs when $0 < a < 1$. Signup and view all the answers

Describe the process of forming exponential rules?

Begin by defining variables for the given situation. Then substitute known information to determine the specific equation for this rule. Ensure any constants match the definitions. Signup and view all the answers

State the formula for the overall amount with compound interest. Define the variables.

The overall amount is given by $A = P(1 + \frac{r}{100})^n$, where ( P ) is principal amount, ( r ) is the rate of interest, per period the interests is added to, and ( n ) is the number of time periods. Signup and view all the answers

Explain why understanding rates of return for the investment period is important?

Understanding rates of return over the investment periods helps you manage the investment with its benefits. For example, knowing that its possible to increase total value to 100% then 200% return, instead of the initial 70%. Signup and view all the answers

How could an investor maximise the rate of return from an asset?

To maximise rate of return, an investor would need to account both the capital value of the asset as well as any income from the asset. Such factors depend on the external market. Signup and view all the answers

Explain in words that 'The logarithm of a number to a given base is the power (or index) to which the base is raised to give the number'.

A logarithm asks, 'To what power must I raise this base to get this number?' The exponent needed is the logarithm. Signup and view all the answers

If $a^x = y$, what are the logarithmic and exponential forms of these equations?

If $a^x = y$, then $y$ is the argument, a is the base and x is the exponential value. $a > 0$, $y > 0$. $x = log_a y$. Thus, log y is read as 'log to the base of a equals x.' Also, 'a' to the power of 'x' equals 'y.' Signup and view all the answers

State three logarithm laws that are similar to the index laws?

The Log law for addition states that $log_a x + log_a y = log_a(xy)$. The Log law for subtraction states that $log_ax - log_a y = log_a(\frac{x}{y})$. The Log law for powers states that $log_a (x^n) = nlog_a x$ Signup and view all the answers

What is the value of $log_e 1$, the natural logarithm of 1? Why?

It equals zero, given all real numbers are able to have zero index, as long as they are not zero themselves. Signup and view all the answers

In solving exponential equations, why is it useful to rewrite in logarithmic forms?

Rewriting in logarithmic form allows us to solve for the exponent, especially when we cannot easily express both sides of the equation with a common base. Signup and view all the answers

Explain the value of $log_{10}A = B$ in terms of A. Explain the value of A in terms of the base 10?

A has the value based on order of magnitude, given $10^B = A$. Signup and view all the answers

When might a logarithmic scale be used?

Logarithmic scales are used to compress a wide range of values into a more manageable and interpretable range. Particularly where data is generated from growth and decay situations. Signup and view all the answers

State when an increase of a number 'x' occurs using an order of magnitude?

To increase a number by an order of magnitude n we multiply x by $10^n$. Signup and view all the answers

How must scientific notation be arranged to measure its order of magnitude?

It must be written in the standard index notation, and the order is measured using the 'power of 10' that relates directly to the power of digits in the standard form. Signup and view all the answers

What can be said about the relationship when variables (linked with exponential relationships) can be represented as a linear relationship using logarithms?

A straight, linear relation links all variables. Thus it is possible to be represented in simpler forms and scales. Signup and view all the answers

Describe how you could find the point of intersection of a simple exponential and a horizontal line?

To do this, draw accurate graphs and read off co-ordinates or use a substitution and equate power in both sides of the equation, with a common base. Signup and view all the answers

What are the axis's of linear relationship charts and graphs are in logarithmic questions?

A logarithmic chart or graph uses the logarithm of quantity on at least one of the axis's. Signup and view all the answers

Flashcards

What is Lmag?

The telescope's light magnitude limit; magnitude of faintest visible star.

What is Gmag?

Telescope's brightness increase capacity