Convert to/from a number

Learning Objectives Convert a logarithmic equation to its equivalent exponential form to solve for an unknown number. Convert an exponential equation to its equivalent logarithmic form to solve for an unknown exponent. Convert a complex number from its rectangular form (a + bi) to its polar form (r(cosθ + isinθ)). Convert a complex number from its polar form to its rectangular form. Solve multi-step equations involving logarithms by applying conversion principles. Determine the modulus (r) and argument (θ) of a complex number to convert it from its standard representation to a numerical representation based on distance and angle. How can you find the exact time it takes for an investment to double at a specific interest rate? 🏦 It involves converting an equation to find a s...

TermDefinitionExample Logarithmic FormA way of expressing an exponent. The equation y = log_b(x) asks, 'What exponent y do I need to raise the base b to in order to get the number x?'log₂(8) = 3 is the logarithmic form of the statement '2 to the power of 3 equals 8'. Exponential FormThe standard way of representing a base raised to a power. The equation x = b^y shows that the number x is the result of the base b raised to the exponent y.2³ = 8 is the exponential form of the statement 'the base-2 logarithm of 8 is 3'. Complex Number (Rectangular Form)A number written in the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part. It corresponds to the point (a, b) on the complex plane.z = 3 + 4i is a complex number with a...

Logarithm-Exponent Equivalence y = \log_b(x) \iff x = b^y This is the fundamental rule for converting between logarithmic and exponential forms. Use it to rewrite an equation into a form that is easier to solve. This is key when the unknown number is 'trapped' inside a logarithm or as an exponent. Rectangular to Polar Conversion For z = a + bi: \quad r = \sqrt{a^2 + b^2} \quad \text{and} \quad \theta = \arctan\left(\frac{b}{a}\right) Use these formulas to find the two numbers (r and θ) that define a complex number in polar form. Be careful to adjust θ to the correct quadrant based on the signs of a and b. Polar to Rectangular Conversion For z = r(\cos\theta + i\sin\theta): \quad a = r\cos\theta \quad \text{and} \quad b = r\sin\theta Use these formulas to fi...