Exponential and Logarithmic Functions
Exponential and logarithmic functions test the relationship between growth and its inverse. Bagrut questions include simplifying log expressions, solving exponential and logarithmic equations, analysing graphs, and applying growth or decay models.
Core rules
- a^x·a^y=a^(x+y), a^x/a^y=a^(x−y), (a^x)^y=a^(xy), a^0=1 for a>0.
- log_a(xy)=log_a x+log_a y, log_a(x/y)=log_a x−log_a y, log_a(x^n)=n log_a x.
- log_a x is defined only for x>0, with a>0 and a≠1.
- Change of base: log_a x=log_b x/log_b a. In particular ln x=log_e x.
- To solve a^(f(x))=a^(g(x)), use f(x)=g(x); to solve different bases, take logarithms of both sides.
- For inequalities, the direction flips when the base is between 0 and 1.
Worked example
Solve 2^(x+1)=5^x. Take natural logs: (x+1)ln2=x ln5. Expand: x ln2+ln2=x ln5. Collect: x(ln5−ln2)=ln2, so x=ln2/(ln5−ln2). This is approximately 0.756. Substituting back shows both sides are equal, and no domain restriction is violated because exponentials are defined for all real x.
Traps
log(a+b) is not log a+log b. Do not cancel log across a sum. Dividing by ln2−ln5 without noticing it is nonzero is fine, but dropping the sign gives the wrong value. In log equations, always check that every solution keeps all arguments positive; otherwise it is extraneous.